Lecture 13: Set operations. Ordered set
1. Union of sets
The union of the sets X and Y is a set consisting of all those and only those elements that belong to at least one of the sets X or Y, i.e. belong to X or belong to Y.
The union of X and Y is denoted by X∪Y
Formally x∈X∪Y ⇔ x∈X or x∈Y
Example 1. If X=(1,2,3,4,5) and Y=(2,4,6,8), then
X∪Y=(1,2,3,4,5,6,7,8)
Example 2. If X=(x:x - ex.gr.), and Y=(x:x - gib.), then
X∪Y=(x:x - either ex., or gib).
Example 3. If X is the set of points on the left circle and Y is the set of points on the right circle, then
X∪Y is the shaded area bounded by both circles.
The concept of unification can be extended to larger number sets, onto a system of sets. Let us denote by M = (X 1 ,X 2 , ...,X n ) a collection of n sets X 1 ,X 2 , ...,X n , sometimes called a system of sets. Union of these sets
∪X i =∪(X∈M), Х=X 1 ∪X 2 ∪...∪X n
is a set consisting of all those and only those elements that belong to at least one of the sets of a given system M.
For combined sets the following are true:
- X∪Y = Y∪X - commutative law
- (X∪Y)∪Z = X∪(Y∪Z) = X∪Y∪Z – associative law,
the validity of which follows from the fact that the left and right sides of the equalities consist of the same elements.
Obviously, X∪∅ = X. From this we can see that ∅ plays the role of zero in set algebra.
2. Intersection of sets
The intersection of sets X and Y is a set consisting of all those and only those elements that belong to both set X and set Y.
The intersection of sets is denoted by X∩Y.
Formally x∈X∩Y ⇔ x∈X and x∈Y
Example 4. X=(1,2,3,4,5) Y=(2,4,6,8) X∩Y = (2,4)
Example 5. If X is the set of points on the left circle and Y is the set of points on the right circle, then X∩Y is the shaded area, which is common part both circles.
Sets X and Y are called disjoint if they do not have common elements, that is, if X∩Y=∅.
Example 7. (1,2,3) and (4,5,6)
Unlike number algebra, where there are three possibilities: a X=Y; X⊂Y; Y⊂X; X∩Y=∅ and X and Y are in general position. The sets X and Y are said to be in general position if three conditions are met: Similar to union, the concept of intersection can be extended to a system of sets: ∩X=∩X i =X 1 ∩X 2 ∩...∩X n The intersection of sets is a set whose elements belong to each of the sets of the system M. For the intersection of sets the following are true: Note also that the relation X∩∅=∅ holds. Example 8. A=(a,b), B=(b,c), C=(a,c). A∩B∩C=∅, although A∩B=(b), B∩C=(c) The set difference is defined for only two sets. The difference of sets X and Y is a set consisting of all those and only those elements that belong to X and do not belong to Y. Denoted by: X\Y. Formally: x∈X\Y ⇔ x∈X and x∉Y Example 9. (see Example 1) X=(1,2,3,4,5), Y=(2,4,6,8), X\Y=(1,3,5), Y\X =(6.8) The difference of sets does not have the property of commutativity. If A\B=∅, then A⊂B - put? back at A∩B≠∅ The role of zero in set algebra is played by the empty set. Is there such a set that plays the role of “1”, i.e. satisfies the condition: X∪I = X, which means that the intersection or “common part” of the set I and the set X for any set X coincides with this set itself. This is possible only if the set I contains all the elements of which the set X can consist, so that any set X is completely contained in the set I. A set I that satisfies this condition is called complete, or universal, or identity. If, in some consideration, only subsets of a certain fixed set are involved, then this largest set will be considered universal and denoted by I. Example 12 (Example 1). I - set of integers Example 13 (Example 2). I - set of students. gr. Example 14 (Example 3). I - sheet of paper, board A universal set is usually denoted graphically as a set of points in a rectangle, and individual sets as separate areas within this rectangle. The representation of sets as regions within a rectangle representing the universal set is called an Euler-Venn diagram. The universal set has an interesting property that has no analogy in ordinary algebra, namely, for any set X the relation X∪I = I holds. The set determined from the relation X¯ = I\X is called the complement of the set X (to the universal set I). In the diagram, the set X¯ represents the unshaded area. Formally: X = (x: x∈I and x∉X). From the definition it follows that X and X¯ do not have common elements. X∩X¯=∅. In addition, there are no elements of I that do not belong to either X or X¯ (its complement), since those elements that do not belong to X belong to X¯ (its complement). Therefore, X∪X¯=I. From the symmetry of this formula with respect to X and X¯ it follows not only that X¯ is the complement of X, but also that X is the complement of X¯. But the complement of X¯ is X¯¯. Thus, X¯ ¯=X¯. Using the addition operation, we represent the difference of sets: X\Y = (x: x∈X and x∉Y) =( x: x∈X and x∈Y¯), i.e. X\Y= X∩Y¯. Order of operations: Parentheses are used to change the order. One of the most common operations on sets is the operation of partitioning a set into a system of subsets. Thus, the system of courses of a given faculty is a division of the many students of the faculty; The group system of this course is a division of the set of students in the course. Example. Products of the enterprise: - premium grade, I, II, defective. Consider some set M and a system of sets M = (X 1, X 2, ..., X n) A system of sets M is called a partition of a set M if it satisfies the following conditions: Any set X from M is a subset of the set M ∀X∈M: X⊆M; Any two sets X and Y from M are disjoint ∀X∈M, ∀Y∈M: X≠Y → X∩Y=∅. The union of all sets included in the partition gives the set M X 1 ∪X 2 ∪...∪ X n =M. Using the operations of union, intersection and addition, various algebraic expressions can be composed from sets. If the algebraic expressions V(X,Y,Z) and S(X,Y,Z) represent the same set, then they can be equated to each other, obtaining an algebraic identity of the form V(X,Y,Z) = S( X,Y,Z) The set of elements belonging to either A or B is called a symmetric difference or disjunctive sum. S = A⊕B = (A\B)∪(B\A) = (A∩B¯)∪(A¯∪B) = (A∪B)∩(A∩B)¯ For the symmetric difference the following laws are satisfied: An ordered set (or tuple) is a sequence of elements, that is, a collection of elements in which each element occupies a specific place. The elements themselves are components of a tuple. Example 1. Many people standing in line, many words in a phrase, alphabet. In all these sets, the place of each element is completely definite and cannot be arbitrarily changed. The number of elements of a tuple is called its length. A tuple is denoted by brackets "< >", sometimes round "()". A= . Tuples of length 2 are called ordered pairs, 3 - triples, n-kami. Special case: tuple of length 1 - tuple of length 0 —< >or ∧ is an empty tuple. The difference between a tuple and an ordinary set: a tuple can have identical elements. We will call ordered sets whose elements are real numbers vectors or points in (n-dimensional) space. Pr 1 = a 1 , Pr 2 = a 2 , Pr i = a i , Pr 1 2 = — two-element tuple. The projection of a tuple onto an empty set of axes is an empty tuple. Generalizing these concepts, we will consider the ordered n-element set of real numbers (a 1, ..., a n) as a point in an imaginary n-dimensional space (sometimes called hyperspace), or as an n-dimensional vector. In this case, we will consider the components of the n-element tuple a as projections of this tuple onto the corresponding axes. Pr i a = a i , i=1,2,...,n Pr i,j,...,l a = , i=1,2,...,n Two vectors are equal if they have the same length and their corresponding coordinates are equal. = ⇔ m = n and a 1 = b 1, b 1 = b 2, ... Components of a tuple (vector) can also be components of tuples (vectors): Example. Words in a sentence The direct (Cartesian) product of the sets X and Y is a set consisting of all those and only those ordered pairs, the first component of which belongs to the set X, and the second belongs to the set Y. Formally: X*Y = ( Example 2. Let X=<1,2>,Y=<1,3,4> Then X*Y=(<1,1>,<1,3>,<1,4>,<2,1>,<2,3>,<2,4>) See fig. A). Example 3. Let X and Y be segments of the real axis. The direct product X*Y is represented by a shaded rectangle. See fig. b). The direct product changes when the order of the factors changes, i.e. The direct product of the sets X 1 , X 2 , ..., X n is a set denoted by X 1 *X 2 *...*X n and consisting of all those and only those tuples of length n whose right component belongs to X 1 , the second - X 2, etc. Obviously X*Y = ∅ ⇔ X = ∅ or Y = ∅. Similarly, X 1 *X 2 *...*X n = ∅ if and only if at least one of the sets X 1 , X 2 , ..., X n is empty. A special case of a direct product is the concept of powers of a (Cartesian) set - the direct product of identical sets M s =M*M*...*M, M 1 =M, M 0 =∧. Usually R is a set of real numbers, then R 2 =R*R is a real plane and R 3 =R*R*R is a three-dimensional real space. Example. A=(a,b,c,d,e,f,g,h), B=(1,2,3, ...,8) Then A*B =(a 1, a 2, a 3, ..., h7, h8) is a set denoting all 64 cells of the chessboard. Example. Let A be a finite set whose elements are symbols (letters, numbers, punctuation marks, etc.). Such sets are usually called alphabets. The elements of the set a n are called words of length n in the alphabet A. The set of all symbols in the alphabet A is the set A * = ∪A i = A 1 ∪A 2 ∪A 3 ... . When writing words, it is not customary to use commas, parentheses, or separators. WORD ⇔<С,Л,О,В,О> Theorem. Let a 1 , a 2 , ..., a n be finite sets and |a 1 | = m 1 , |a 2 |=m 2 , ..., |a n |=m n . Then the power of the set a 1 *a 2 *a 3 *...*a n is equal to the product of the powers a 1 , a 2 , ..., a n |a 1 *a 2 *...*a n |=|a 1 |*|a 2 |*|a 3 |*...*|a n |= m 1 *m 2 *...*m n Corollary |a n |=|A| n The operation of programming a set is closely related to the operation of designing a tuple and can only be applied to sets whose elements are tuples of the same length. Let M be a set consisting of tuples of length S. Then the proline of a set M will be the set of prolines of all tuples from M Example. Let M=(<1,2,3,4,5>,<2,1,3,5,5>,<3,3,3,3,3>,<3,2,3,4,3>} then Pr 2 M=(2,1,3), Pr 3 M=(3), Pr 4 M=(4,5,3), Pr 24 M=(<2,4>,<1,5>,<3,3>), Pr 13 M=(<1,3>,<2,3>,<3,3>), Pr 15 M=(<1,5>,<2,5>,<1,3>), Pr 25 M=(<2,5>,<1,5>,<3,3>,<2,3>}. It is obvious that if M=X*Y then Pr 1 M=X, Pr 2 M=Y and if Q⊆Х*Y then Pr 1 Q⊆Х and Pr 2 Q⊆Y Example. V=( Pr 1 V=(a,c,d) Pr 1 2V=( Pr 2 3V=( Pr 1 3V=( Let V be a set of vectors of the same length S. Pr i V = (Pr i v/v∈Y), Pr i i ...i k v = ( Pr i i ...i k v/v∈Y). If V =A 1 *A 2 *...*A n , then Pr i i ...i k V=A i1 *A i2 *...*A ik . In the general case, Pr i V is not necessarily a direct product: it can be a subset. Solving some mathematical problems forces you to find intersection and union of number sets. We have already become acquainted with the accepted notation for numerical sets, and in this article we will carefully and with examples understand how to find the intersection and union number sets. These skills will be useful, in particular, in the process solutions to inequalities with one variable and their systems. Page navigation. By the simplest cases we will mean finding the intersection and union of numerical sets, which are a set of individual numbers. In these cases it is sufficient to use definitions of intersection and union of sets. Let us remind you that Definition.
unification two sets is a set, each element of which is an element of any of the original sets, and intersection sets is a set consisting of all common elements of the original sets. From these definitions it is easy to obtain following rules finding the intersection and union of sets: Indeed, the set obtained by the first rule will consist of all elements belonging to at least one of the original sets, and therefore will be a union of these sets by definition. And the set compiled according to the second rule will contain everything common elements of the original sets, that is, will be the intersection of the original sets. Let's look at specific examples application of the stated rules to find the intersection and union of sets. For example, let's say we need to find the union of number sets A=(3, 5, 7, 12) and B=(2, 5, 8, 11, 12, 13) . We write down all the elements, for example, of set A, we have 3, 5, 7, 12, and to them we add the missing elements of set B, that is, 2, 8, 11 and 13, as a result we have the numerical set (3, 5, 7, 12, 2, 8, 11, 13) . It doesn’t hurt to order the elements of the resulting set; as a result, we get the desired union: A∪B=(2, 3, 5, 7, 8, 11, 12, 13). Now let's find the intersection of two numerical sets from the previous example A=(3, 5, 7, 12) and B=(2, 5, 8, 11, 12, 13). According to the rule, we will sequentially go through the elements of the first set A and check whether they are included in set B. We take the first element 3, it does not belong to the set B, therefore, it will not be an element of the desired intersection. Let's take the second element of set A, this is the number 5. It belongs to the set B, therefore it also belongs to the intersection of the sets A and B. This is how the first element of the desired intersection is found - the number 5. Let's move on to the third element of set A, this is the number 7. It does not belong to B, which means it does not belong to the intersection. Finally, the last element of set A remains - the number 12. It belongs to the set B, therefore, it is also an intersection element. So, the intersection of the sets A=(3, 5, 7, 12) and B=(2, 5, 8, 11, 12, 13) is a set consisting of two elements 5 and 12, that is, A∩B =(5, 12) . As you noticed, above we talked about finding the intersection and union of two numerical sets. As for the intersection and union of three or more sets, finding it can be reduced to sequentially finding the intersection and union of two sets. For example, to find the intersection of three sets A, B and D, you can first find the intersection of A and B, and then find the intersection of the resulting result with the set D. And now specifically: let’s take the number sets A=(3, 9, 4, 3, 5, 21), B=(2, 7, 9, 21) and D=(7, 9, 1, 3) and find their intersection . We have A∩B=(9, 21) , and the intersection of the resulting set with set D is (9) . Thus, A∩B∩D=(9) . However, in practice, to find the intersection of three, four, etc. For the simplest numerical sets, consisting of a finite number of individual numbers, it is convenient to use rules similar to the rules indicated above. So, to obtain a union of three or more sets specified type, we need to add the missing numbers of the second to the numbers of the first numerical set, add the missing numbers of the third set to the written numbers, and so on. To clarify this point, let’s take the number sets A=(1, 2) , B=(2, 3) and D=(1, 3, 4, 5) . To elements 1 and 2 of the numerical set A we add the missing number 3 of the set B, we get 1, 2, 3, and to these numbers we add the missing numbers 4 and 5 of the set D, as a result we get the union of three sets we need: A∪B∪C= (1, 2, 3, 4, 5) . As for finding the intersection of three, four, etc. numerical sets consisting of a finite number of individual numbers, you need to sequentially go through the numbers of the first set and check whether the number being checked belongs to each of the remaining sets. If yes, then this number is an element of intersection, if not, then it is not. Here we just note that it is advisable to take the set with the smallest number elements. As an example, let's take four numerical sets A=(3, 1, 7, 12, 5, 2) , B=(1, 0, 2, 12) , D=(7, 11, 2, 1, 6) , E =(1, 7, 15, 8, 2, 6) and find their intersection. Obviously, set B contains the fewest elements, so to find the intersection of the original four sets, we will take the elements of set B and check whether they are included in the remaining sets. So, we take 1, this number is elements of both sets A, and D and E, so this is the first element of the desired intersection. Let's take the second element of set B - it's zero. This number is not an element of the set A, so it will not be an element of the intersection. We check the third element of set B – the number 2. This number is an element of all other sets, therefore, it is the second intersection element found. Finally, the fourth element of the set B remains. This number is 12, it is not an element of the set D, therefore, it is not an element of the desired intersection. As a result, we have A∩B∩D∩E=(1, 2) . In our example we have records AND Next, another coordinate line is drawn; it is convenient to place it under the existing ones. It will display the desired intersection or union. All boundary points of the original numerical sets are marked on this coordinate line. In this case, these points are first marked with dashes; later, when the nature of the points with these coordinates is clarified, the dashes will be replaced by punctured or non-punctured points. In our case, these are points with coordinates −3 and 7. The points depicted on the lower coordinate line at the previous step of the algorithm allow us to consider the coordinate line as a set numerical intervals and points, which we talked about in. In our case, we consider the coordinate line as a set of the following five numerical sets: (−∞, −3) , (−3) , (−3, 7) , (7) , (7, +∞) . And all that remains is to check, one by one, whether each of the written sets is included in the desired intersection or union. All conclusions drawn are marked step by step on the lower coordinate line: if the interval is included in the intersection or union, then a hatch is drawn above it, if the point is included in the intersection or union, then the stroke denoting it is replaced with a solid point, if it is not included, then we make it punctured. In this case, the following rules should be adhered to: Simply put, the intersection of the number sets A and B is the union of all the number intervals of the sets A and B that are simultaneously hatched, and all the individual points that belong to both A and B at the same time. And the union of two numerical sets is the union of all numerical intervals over which at least one of the sets A or B has shading, as well as all unpunctured individual points. Let's return to our example. Let's finish finding the intersection of sets. To do this, we will sequentially check the sets (−∞, −3) , (−3) , (−3, 7) , (7) , (7, +∞) . We start with (−∞, −3), for clarity we highlight it in the drawing: We do not include this gap in the required intersection, since it is not included in either A or B (there is no shading above this gap). So at this step we do not mark anything in our drawing and it retains its initial appearance: Let's move on to the next set (−3). The number −3 belongs to the set B (this is a non-punctured point), but obviously does not belong to the set A, therefore it does not belong to the desired intersection. Therefore, on the lower coordinate line we make a point with coordinate −3 punctured: We check the following set (−3, 7) . It is included in set B (there is a hatch above this interval), but is not included in set A (there is no hatch above this interval), therefore, it will not be included in the intersection. Therefore, we do not mark anything on the lower coordinate line: Let's move on to set (7). It is included in set B (the point with coordinate 7 is an interior point of the interval [−3, +∞)), but not included in set A (this point is punctured), so it will not be included in the desired intersection. Mark the point with coordinate 7 as punctured: It remains to check the interval (7, +∞) . It is included in both set A and set B (there is a hatching above this gap), therefore it is also included in the intersection. We put shading over this gap: As a result, on the lower coordinate line we received an image of the desired intersection of the sets A=(7, +∞) and B=[−3, +∞) . Obviously, it represents the set of all real numbers, greater than seven, that is, A∩B=(7, +∞) . Now let's find the union of sets A and B. We begin a sequential check of the sets (−∞, −3) , (−3) , (−3, 7) , (7) , (7, +∞) for their inclusion in the desired union of two numerical sets A and B . The first set (−∞, −3) is not included in either A or B (there is no shading above this interval), so this set will not be included in the desired union: The set (−3) is included in the set B, therefore it will also be included in the union of the sets A and B: The interval (−3, 7) is also included in B (there is a hatch above this interval), therefore it will be integral part the desired union: Set (7) will also be included in the desired union, since it is included in the numerical set B: Finally, (7, +∞) is included in both set A and set B, therefore, it will also be included in the desired union: Based on the resulting image of the union of sets A and B, we conclude that A∩B=[−3, +∞) . Having gained some practical experience, checking the inclusion of individual intervals and numbers in the intersection or union can be done orally. Thanks to this, you can record the result very quickly. Let's show what the solution to the example will look like if we don't give an explanation. Example. Find the intersection and union of sets A=(−∞, −15)∪(−5)∪∪(12) And B=(−20, −10)∪(−5)∪(2, 3)∪(17). Solution. Let us depict these numerical sets on coordinate lines, this will allow us to obtain images of their intersection and union: Answer: A∩B=(−20, −15)∪(−5)∪(2, 3) And A∪B=(−∞, −10)∪(−5)∪∪(12, 17). It is clear that with proper understanding, the algorithm outlined above can be optimized. For example, when finding the intersection of sets, there is no need to check all the intervals and sets consisting of individual numbers into which the boundary points of the original sets are divided into the coordinate line. You can limit yourself to checking only those intervals and numbers that make up the set A or B. The remaining intervals will still not be included in the intersection, since they do not belong to one of the original sets. Let us illustrate this by analyzing the solution to the example. Example. What is the intersection of the number sets A=(−2)∪(1, 5) and B=[−4, 3]? Solution. Let's construct geometric images of number sets A and B: Boundary points given sets divide the number line into the following sets: (−∞, −4) , (−4) , (−4, −2) , (−2) , (−2, 1) , (1) , (1, 3) , (3) , (3, 5) , (5) , (5, +∞) . It is easy to see that the numerical set A can be “assembled” from the sets just written by combining (−2) , (1, 3) , (3) and (3, 5) . To find the intersection of sets A and B, it is enough to check whether the latter sets are included in set B. Those of them that are included in B will constitute the desired intersection. Let's perform the appropriate check. Obviously, (−2) is included in the set B (since the point with coordinate −2 is an interior point of the segment [−4, 3]). The interval (1, 3) is also included in B (there is a hatch above it). Set (3) is also included in B (the point with coordinate 3 is a boundary and non-punctured point of the set B). And the interval (3, 5) is not included in the numerical set B (there is no shading above it). Having marked the conclusions made on the drawing, it will take this form Thus, the desired intersection of two original numerical sets A and B is the union of the following sets (−2) , (1, 3) , (3) , which can be written as (−2)∪(1, 3] . Answer: {−2}∪(1, 3]
. All that remains is to discuss how to find the intersection and union of three and more number sets. This problem can be reduced to sequentially finding the intersection and union of two sets: first the first with the second, then the obtained result with the third, then the obtained result with the fourth, and so on. Or you can use an algorithm similar to the one already announced. Its only difference is that checking the occurrence of intervals and sets consisting of individual numbers must be carried out not by two, but by all initial sets. Let's consider an example of finding the intersection and union of three sets. Example. Find the intersection and union of three number sets A=(−∞, 12] , B=(−3, 25] , D=(−∞, 25)∪(40) . Solution. First, as usual, we depict numerical sets on coordinate lines, and to the left of them we place a curly bracket indicating intersection and a square bracket for union, and below we depict coordinate lines with the boundary points of numerical sets marked with strokes: So the coordinate line turns out to be represented by numerical sets (−∞, −3) , (−3) , (−3, 12) , (12) , (12, 25) , (25) , (25, 40) , (40) , (40, ∞) . We begin searching for intersections; to do this, we look in turn to see if the recorded sets are included in each of the sets A, B and D. All three initial numerical sets include the interval (−3, 12) and the set (12) . They constitute the desired intersection of the sets A, B and D. We have A∩B∩D=(−3, 12] . In turn, the desired union will consist of the sets (−∞, −3) (included in A), (−3) (included in A), (−3, 12) (included in A), (12) (included in A ), (12, 25) (included in B ), (25) (included in B ) and (40) (included in D ). Thus, A∪B∪D=(−∞, 25]∪(40) . Answer: A∩B∩D=(−3, 12] , A∪B∪D=(−∞, 25]∪(40) . In conclusion, we note that the intersection of numerical sets is often empty set. This corresponds to cases when the original sets do not have elements that simultaneously belong to all of them. (10, 27) , (27) , (27, +∞) . None of the written sets is simultaneously included in the four original sets, which means that the intersection of the sets A, B, D and E is the empty set. Answer: A∩B∩D∩E=∅. Bibliography. Solution of some mathematical problems involves finding the intersection and union of numerical sets. In the article below we will consider these actions in detail, including specific examples. The acquired skill will be applicable to solving inequalities with one variable and systems of inequalities. When we talk about the simplest cases in the topic under consideration, we mean finding the intersection and union of numerical sets, which are a set of individual numbers. In such cases, it will be sufficient to use the definition of intersection and union of sets. Definition 1 Union of two sets is a set in which each element is an element of one of the original sets. Intersection of many is a set that consists of all common elements of the original sets. From the above definitions The following rules logically follow: To form a union of two numerical sets with a finite number of elements, it is necessary to write down all the elements of one set and add to them the missing elements from the second set; To create the intersection of two numerical sets, it is necessary to check the elements of the first set one by one to see if they belong to the second set. Those of them that turn out to belong to both sets will constitute the intersection. The set obtained according to the first rule will include all elements belonging to at least one of the original sets, i.e. will become the union of these sets by definition. The set obtained according to the second rule will include all the common elements of the original sets, i.e. will become the intersection of the original sets. Let's consider the application of the resulting rules using practical examples. Example 1 Initial data: numerical sets A = (3, 5, 7, 12) and B = (2, 5, 8, 11, 12, 13). It is necessary to find the union and intersection of the original sets. Solution
Answer: union of the original sets – A ∪ B = (2, 3, 5, 7, 8, 11, 12, 13); intersection of the original sets - A ∩ B = (5, 12). All of the above applies to working with two sets. As for finding the intersection and union of three or more sets, the solution to this problem can be reduced to sequentially finding the intersection and union of two sets. For example, to determine the intersection of three sets A, B, and C, it is possible to first determine the intersection of A and B, and then find the intersection of the resulting result with the set C. Using an example, it looks like this: let the numerical sets be given: A = (3, 9, 4, 3, 5, 21), B = (2, 7, 9, 21) and C = (7, 9, 1, 3 ) . The intersection of the first two sets will be: A ∩ B = (9, 21), and the intersection of the resulting set with the set A ∩ B = (9, 21). As a result: A ∩ B ∩ C = ( 9 ) . However, in practice, in order to find the union and intersection of three or more simple numerical sets that consist of a finite number of individual numbers, it is more convenient to apply rules similar to those indicated above. That is, to find a union of three or more sets of the specified type, it is necessary to add the missing elements of the second set to the elements of the first set, then the third, etc. For clarification, let's take numerical sets: A = (1, 2), B = (2, 3), C = (1, 3, 4, 5). The number 3 from set B will be added to the elements of the first set A, and then the missing numbers 4 and 5 from set C. Thus, the union of the original sets: A ∪ B ∪ C = (1, 2, 3, 4, 5). As for solving the problem of finding the intersection of three or more numerical sets that consist of a finite number of individual numbers, it is necessary to go through the numbers of the first set one by one and step by step check whether the number in question belongs to each of the remaining sets. For clarification, consider number sets: A = (3, 1, 7, 12, 5, 2) B = (1, 0, 2, 12) C = (7, 11, 2, 1, 6) D = (1, 7, 15, 8, 2, 6). Let's find the intersection of the original sets. Obviously, set B has the fewest elements, so these are the ones we will check to determine whether they are included in the remaining sets. Number 1 of set B is an element of other sets, and therefore is the first element of the desired intersection. The second number of set B - number 0 - is not an element of set A, and, therefore, will not become an element of intersection. We continue checking: number 2 of set B is an element of other sets and becomes another part of the intersection. Finally, the last element of set B - the number 12 - is not an element of set D and is not an element of intersection. Thus, we get: A ∩ B ∩ C ∩ D = ( 1 , 2 ) . Let's mark an arbitrary point on the coordinate line, for example, with coordinates - 5, 4. The specified point will divide the coordinate line into two numerical intervals - two open rays (-∞, -5,4) and (-5,4, +∞) and the point itself. It is easy to see that, in accordance with the definition of a union of sets, any real number will belong to the union (- ∞, - 5, 4) ∪ (- 5, 4) ∪ (- 5, 4, + ∞). Those. the set of all real numbers R = (- ∞ ; + ∞) can be represented in the form of the union obtained above. Conversely, the resulting union will be the set of all real numbers. Let us note that for this point it is possible to attach to any of the open beams, then it will become simple numerical beam(- ∞ , - 5 , 4 ] or [ - 5 , 4 , + ∞) . In this case, the set R will be described by the following unions: (- ∞ , - 5 , 4 ] ∪ (- 5 , 4 , + ∞) or (- ∞ , - 5 , 4) ∪ [ - 5 , 4 , + ∞). . Similar reasoning is valid not only with respect to a point on a coordinate line, but also with respect to a point on any numerical interval. That is, if we take any internal point of any arbitrary interval, it can be represented as a union of its parts obtained after division given point, and the point itself. For example, a half-interval (7, 32] and a point 13 belonging to this numerical interval are given. Then the given half-interval can be represented as a union (7, 13) ∪ (13) ∪ (13, 32] and vice versa. We can include the number 13 in any of the intervals and then the given set (7, 32 ] can be represented as (7, 13 ] ∪ (13, 32 ] or (7, 13 ] ∪ (13, 32 ]. We can also take not the internal point of a given half-interval, and its end (the point with coordinate 32), then the given half-interval can be represented as the union of the interval (7, 32) and a set of one element (32). Thus: (7, 32] = (7, 32) ∪ ( 32 ) . Another option: when not one, but several points are taken on a coordinate line or a numerical interval. These points will divide the coordinate line or numerical interval into several numerical intervals, and the union of these intervals will form the original sets. For example, points on the coordinate line are given with coordinates - 6, 0, 8, which will divide it into intervals: (- ∞, - 6), (- 6, 0), (0, 8), (8, + ∞) . In this case, the set of all real numbers, the embodiment of which is the coordinate line, can be represented as a combination of the resulting intervals and the indicated numbers: (- ∞ , - 6) ∪ { - 6 } ∪ (- 6 , 0) ∪ { 0 } ∪ (0 , 8) ∪ { 8 } ∪ (8 , + ∞) . The topic of finding the intersection and union of sets can be clearly understood if you use images of given sets on a coordinate line (unless we are talking about the simplest cases discussed at the very beginning of the article). We'll consider general approach, which allows you to determine the result of the intersection and union of two numerical sets. Let us describe the approach in the form of an algorithm. We will consider its steps gradually, each time citing the next stage of solving a specific example. Example 2 Initial data: given numerical sets A = (7, + ∞) and B = [ - 3, + ∞). It is necessary to find the intersection and union of these sets. Solution
In our example, to write the intersection and union of numerical sets we have: and Let's draw another coordinate line, placing it under the existing ones. It will be needed to display the desired intersection or union. On this coordinate line, all the boundary points of the original numerical sets are marked: first with dashes, and later, after clarifying the nature of the points with these coordinates, the dashes will be replaced by punctured or non-punctured points. In our example, these are points with coordinates - 3 and 7. The points that are depicted on the lower coordinate line in the previous step of the algorithm make it possible to consider the coordinate line as a set of numerical intervals and points (we talked about this above). In our example, we represent the coordinate line as a set of five numerical sets: (- ∞, - 3), (- 3), (- 3, 7), (7), (7, + ∞). Now you need to check one by one whether each of the written sets belongs to the desired intersection or union. The resulting conclusions are marked in stages on the lower coordinate line: when the gap is part of an intersection or union, a hatch is drawn above it. When a point enters an intersection or union, the stroke is replaced by a solid point; if the point is not part of the intersection or union, it is punctured. In these actions you must adhere to the following rules: A gap becomes part of the intersection if it is simultaneously part of set A and set B (or in other words, if there is shading above this gap on both coordinate lines representing sets A and B); A point becomes part of the intersection if it is simultaneously part of each of the sets A and B (in other words, if the point is a non-punctured or internal point of any interval of both numerical sets A and B); A gap becomes part of a union if it is part of at least one of the sets A or B (in other words, if there is shading over this gap on at least one of the coordinate lines representing the sets A and B. A point becomes part of a union if it is part of at least one of the sets A and B (in other words, the point is a non-punctured or interior point of any interval of at least one of the sets A and B). Briefly summarizing: the intersection of numerical sets A and B is the intersection of all numerical intervals of sets A and B, over which shading is simultaneously present, and all individual points belonging to both set A and set B. The union of numerical sets A and B is the union of all numerical intervals , over which at least one of the sets A or B has shading, as well as all unpunctured individual points. This gap will not be included in the intersection because it is not part of either set A or set B (no shading). And so our drawing retains its original appearance: Consider the following set (-3). The number - 3 is part of set B (not a punctured point), but is not part of set A, and therefore will not become part of the desired intersection. Accordingly, on the lower coordinate line we make a point with coordinate - 3: We evaluate the following set (- 3, 7). It is part of set B (there is shading above the interval), but is not included in set A (there is no shading above the interval): it will not be included in the desired intersection, which means that no new marks appear on the lower coordinate line: The next set to check is (7). It is part of the set B (the point with coordinate 7 is an internal point of the interval [ - 3, + ∞)), but is not part of the set A (punctured point), thus, the interval in question will not become part of the desired intersection. Let us mark the point with the coordinate 7 as punched out: And finally, we check the remaining gap (7, + ∞). The gap is included in both sets A and B (hatching is present above the gap), therefore, it becomes part of the intersection. We shade the place above the considered gap: Ultimately, an image of the desired intersection of the given sets was formed on the lower coordinate line. Obviously, it is the set of all real numbers greater than the number 7, i.e.: A ∩ B = (7, + ∞). The first set (- ∞, - 3) is not part of any of the original sets A and B (there are no shadings above the intervals), therefore, the set (- ∞, - 3) will not be included in the desired union: The set ( - 3) is included in the set B, which means it will be included in the desired union of the sets A and B: The set (- 3, 7) is an integral part of set B (there is shading above the interval) and becomes an element of the union of sets A and B: The set 7 is included in the numerical set B, therefore it will also be included in the desired union: The set (7, + ∞), being an element of both sets A and B at the same time, becomes another part of the desired union: Based on the final image of the union of the original sets A and B, we obtain: A ∩ B = [ - 3 , + ∞) . Having some practical experience in applying the rules for finding intersections and unions of sets, the described checks are easily carried out orally, which allows you to quickly write down final result. We will demonstrate at practical example what his solution looks like without detailed explanations. Example 3 Initial data: sets A = (- ∞ , - 15) ∪ ( - 5 ) ∪ [ 0 , 7) ∪ ( 12 ) and B = (- 20 , - 10) ∪ ( - 5 ) ∪ (2 , 3) ∪ (17). It is necessary to determine the intersection and union of the given sets. Solution
Let us mark the given numerical sets on the coordinate lines in order to be able to obtain an illustration of the required intersection and union: Answer: A ∩ B = (- 20, - 15) ∪ (- 5) ∪ (2, 3); A ∪ B = (- ∞ , - 10) ∪ ( - 5 ) ∪ [ 0 , 7 ] ∪ ( 12 , 17 ) . It is also clear that with sufficient understanding of the process, the specified algorithm can be optimized. For example, in the process of finding the intersection, you don’t have to waste time checking all the intervals and sets that represent individual numbers, limiting yourself to considering only those intervals and numbers that make up the set A or B. Other intervals will not be included in the intersection in any case, i.e. To. are not part of the original sets. Let's illustrate what has been said using a practical example. Example 4 Initial data: sets A = ( - 2 ) ∪ [ 1 , 5 ] and B = [ - 4 , 3 ] . It is necessary to determine the intersection of the original sets. Solution
Let us represent the numerical sets A and B geometrically: The boundary points of the original sets will divide the number line into several sets: (- ∞ , - 4) , { - 4 } , (- 4 , - 2) , { - 2 } , (- 2 , - 1) , { 1 } , (1 , 3) , { 3 } , (3 , 5) , { 5 } , (5 , + ∞) . It is easy to see that the numerical set A can be written by combining some of the listed sets, namely: ( - 2), (1, 3), (3) and (3, 5). It will be enough to check these sets for their inclusion also in set B in order to find the desired intersection. Those that will be included in set B and become elements of intersection. Let's check. It is absolutely clear that ( - 2) is part of the set B, because the point with coordinate - 2 is an internal point of the segment [ - 4, 3). The interval (1, 3) and the set (3) are also included in set B (there is a shading above the interval, and the point with coordinate 3 is boundary and not punctured for set B). The set (3, 5) will not be an intersection element, because is not included in set B (there is no shading above it). Let's note all of the above in the drawing: As a result, the desired intersection of two given sets will be the union of sets, which we will write as follows: ( - 2 ) ∪ (1 , 3 ] . Answer: A ∩ B = ( - 2 ) ∪ (1 , 3 ] . At the end of the article, we will also discuss how to solve the problem of finding the intersection and union of several sets (more than 2). Let us reduce it, as recommended earlier, to the need to determine the intersection and union of the first two sets, then the resulting result with the third set, and so on. Or you can use the algorithm described above with the only difference that checking the occurrence of intervals and sets that represent individual numbers must be carried out not by two, but by all given sets. Let's look at an example. Example 5 Initial data: sets A = (- ∞, 12], B = (- 3, 25], D = (- ∞, 25) ꓴ (40). It is necessary to determine the intersection and union of the given sets. Solution
We display the given numerical sets on coordinate lines and place a curly bracket on the left side of them, denoting intersection, as well as a square bracket, denoting union. Below we display coordinate lines with boundary points of numerical sets marked with strokes: Thus, the coordinate line is represented by the following sets: (- ∞, - 3), (- 3), (- 3, 12), (12), (12, 25), (25), (25, 40), ( 40 ) , (40 , + ∞) . We begin to look for intersections, alternately checking the written sets to see if they belong to each of the original ones. All three given sets include the interval (- 3, 12) and the set (- 12): they will become the elements of the desired intersection. Thus, we get: A ∩ B ∩ D = (- 3 , 12 ] . The union of the given sets will make up the following sets: (- ∞ , - 3) - element of set A; ( - 3 ) – element of set A; (- 3, 12) – element of set A; ( 12 ) – element of set A; (12, 25) – element of set B; (25) is an element of set B and (40) is an element of set D. Thus, we get: A ∪ B ∪ D = (- ∞ , 25 ] ∪ ( 40 ) . Answer: A ∩ B ∩ D = (- 3, 12 ]; A ∪ B ∪ D = (- ∞, 25 ] ∪ ( 40 ) . Note also that the desired intersection of numerical sets is often the empty set. This happens in cases where the given sets do not include elements that simultaneously belong to all of them. Example 6 Initial data: A = [ - 7, 7 ]; B = ( - 15 ) ∪ [ - 12 , 0) ∪ ( 5 ) ; D = [ - 15 , - 10 ] ∪ [ 10 , + ∞) ; E = (0, 27). Determine the intersection of given sets. Solution
Let us display the original sets on coordinate lines and the boundary points of these sets on the additional line with strokes. The marked points will divide the number line into sets: (- ∞ , - 15) , ( - 15 ) , (- 15 , - 12) , ( - 12 ) , (- 12 , - 10) , ( - 10 ) , (- 10 , - 7) , ( - 7 ) , ( - 7 , 0) , ( 0 ) , (0 , 5) , ( 5 ) , (5 , 7) , ( 7 ) , (7 , 10) , ( 10 ) , (10, 27) , (27) , (27, + ∞) . None of them is simultaneously an element of all the original sets; therefore, the intersection of the given sets is the empty set. Answer: A ∩ B ∩ D ∩ E = Ø. It is convenient to represent sets in the form of circles, which are called Euler circles. In the figure, the intersection set of the sets X and Y is colored orange. If you notice an error in the text, please highlight it and press Ctrl+Enter Lesson Objectives: “Multiple is many things that we think of as one”
(founder of set theory - Georg Cantor). CANTOR Georg (1845-1918) - German mathematician, logician, theologian, creator of the theory of transfinite (infinite) sets, which had a decisive influence on the development of mathematical sciences at the turn of the 19th and 20th centuries. Set is one of the basic concepts modern mathematics, used in almost all of its sections. Unfortunately, the basic concept of the theory—the concept of set—cannot be given a strict definition. Of course, we can say that a set is a “set”, “collection”, “ensemble”, “collection”, “family”, “system”, “class”, etc. however, all this would not be mathematical definition, but rather the abuse of the vocabulary wealth of the Russian language. In order to define any concept, it is necessary, first of all, to indicate which particular case is more general concept, it is, for the concept of a set this is impossible, because there is no more general concept than a set in mathematics. Often we have to talk about several things united by some characteristic. So, we can talk about the set of all chairs in the room, the set of all cells human body, about the set of all potatoes in a given bag, about the set of all fish in the ocean, about the set of all squares on a plane, about the set of all points on a given circle, etc. The objects that make up a given set are called its elements. For example, many days of the week consist of the elements: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. Many months - from the elements: January, February, March, April, May, June, July, August, September, October, November, December. A bunch of arithmetic operations- from the elements: addition, subtraction, multiplication, division. For example, if A means the set of all natural numbers, then 6 belongs to A, and 3 does not belong to A. If A is the set of all months of the year, then May belongs to A, but Wednesday does not belong to A. If a set contains a finite number of elements, then it is called finite, and if it has infinitely many elements, then it is called infinite. So the set of trees in a forest is finite, but the set of points on a circle is infinite. Paradox in logic- this is a contradiction that has the status of a logically correct conclusion and, at the same time, represents reasoning leading to mutually exclusive conclusions. As already mentioned, the concept of set is at the core of mathematics. Using the simplest sets and various mathematical constructions, you can construct almost any mathematical object. The idea of constructing all mathematics on the basis of set theory was actively promoted by G. Cantor. However, for all its simplicity, the concept of set is fraught with the danger of contradictions or, as they also say, paradoxes. The appearance of paradoxes is due to the fact that not all constructions and not all sets can be considered. The simplest of paradoxes is " barber paradox". One soldier was ordered to shave those and only those soldiers in his platoon who did not shave themselves. Failure to comply with an order in the army, as is known, is a grave crime. However, the question arose whether this soldier should shave himself. If he shave, then he should be classified among the many soldiers who shave themselves, and he has no right to shave such people. If he does not shave himself, he will end up among many soldiers who do not shave themselves, and according to the order he is obliged to shave such soldiers. Paradox. Over sets, as well as over many others mathematical objects, you can perform various operations, which are sometimes called set-theoretic operations or set operations. As a result of operations, new sets are obtained from the original sets. Sets are designated by capital letters with Latin letters, and their elements are lowercase. Record a
R means that the element A belongs to the set R, that is A R. IN otherwise, When A does not belong to the set R, they write a
R . Two sets A And IN are called equal
(A =IN), if they consist of the same elements, that is, each element of the set A is an element of the set IN and vice versa, each element of the set IN is an element of the set A . Comparison of sets. A set A is contained in a set B (a set B includes a set A) if every element of A is an element of B: They say that there are many A contained in many IN or many A is subset
sets IN(in this case they write A IN), if each element of the set A is at the same time an element of the set IN. This dependence between sets is called turning on
. For any set A inclusions occur: Ø A And A A In this case A called subset B, B - superset A. If , then A called own subset IN. notice, that ,
A-priory , The two sets are called equal, if they are subsets of each other Set Operations Intersection. An association. Properties. 1. The operation of combining sets is commutative 2. The operation of combining sets is transitive 3. The empty set X is a neutral element of the set union operation 1. Let A = (1,2,3,4),B = (3,4,5,6,7). Then 2. A = (2,4,6,8,10), B = (3,6,9,12). Let's find the union and intersection of these sets: {2,4,6,8, 10,3,6,9,12}, = {6}. 3. The set of children is a subset of the entire population 4. The intersection of the set of integers with the set of positive numbers is the set of natural numbers. 5. By combining the set of rational numbers with the set irrational numbers is the set of positive numbers. 6. Zero is the complement of the set of natural numbers relative to the set of non-negative integers. Venn diagrams(Venn diagrams) - common name a number of visualization methods and graphic illustration methods widely used in various areas science and mathematics: set theory, proper "Venn diagram" shows all possible relationships between sets or events from a certain family; varieties Venn diagram serve: Euler diagrams, Venn diagram of four sets. Actually "Venn diagram" shows all possible relationships between sets or events from a certain family. A typical Venn diagram has three sets. Venn himself tried to find elegant way with symmetrical figures
, representing a larger number of sets in the diagram, but he was only able to do this for four sets (see figure on the right) using ellipses. Euler diagrams Euler diagrams are similar to Venn diagrams. Euler diagrams can be used to evaluate the plausibility of set-theoretic identities. Task 1. There are 30 people in the class, each of whom sings or dances. It is known that 17 people sing, and 19 people can dance. How many people sing and dance at the same time? Solution: First, let's note that out of 30 people, 30 - 17 = 13 people cannot sing. They all know how to dance, because... According to the condition, each student in the class sings or dances. In total, 19 people can dance, 13 of them cannot sing, which means that 19-13 = 6 people can dance and sing at the same time. Problems involving the intersection and union of sets. Mathematical analysis is the branch of mathematics that deals with the study of functions based on the idea of an infinitesimal function. Basic concepts mathematical analysis are magnitude, set, function, infinite small function, limit, derivative, integral. Size Anything that can be measured and expressed by number is called. Many is a collection of some elements united by some common feature. Elements of a set can be numbers, figures, objects, concepts, etc. The sets are denoted in capital letters, and there are many elements lowercase letters. Elements of sets are enclosed in curly braces. If element x belongs to the set X, then write x∈X (∈
- belongs). A set can be defined in one of two ways: by enumeration and by using a defining property. The set (-∞;+∞) is called number line, and any number is a point on this line. Let a - arbitrary point number line andδ - positive number. The interval (a-δ; a+δ) is called δ-neighborhood of point a. A set X is bounded from above (from below) if there is a number c such that for any x ∈ X the inequality x≤с (x≥c) holds. The number c in this case is called top (bottom) edge set X. A set bounded both above and below is called limited. The smallest (largest) of the upper (lower) faces of a set is called exact top (bottom) edge of this multitude. A bunch of rational numbers. In addition to whole numbers, there are also fractions. A fraction is an expression of the form where p- integer, q- natural. Decimal fractions can also be written as . For example: 0.25 = 25/100 = 1/4. Integers can also be written as . For example, in the form of a fraction with the denominator “one”: 2 = 2/1. So any rational number can be written down decimal- finite or infinitely periodic. Plenty of everyone real numbers. Irrational numbers are infinite non-periodic fractions. These include: Together, two sets (rational and irrational numbers) form the set of real (or real) numbers. If a set does not contain a single element, then it is called empty set and is recorded Ø
. Notation ∀x: |x|<2 → x 2 < 4 означает: для каждого x такого, что |x|<2, выполняется неравенство x 2 < 4. Quantifiers are often used when writing mathematical expressions. Quantifier is called a logical symbol that characterizes the elements following it in quantitative terms. Two sets A and B are equal(A=B) if they consist of the same elements. By union (sum) sets A and B is a set A ∪ B whose elements belong to at least one of these sets. By intersection (product) sets A and B is called a set A ∩ B, the elements of which belong to both the set A and the set B. By difference The sets A and B are called the set AB, the elements of which belong to the set A, but do not belong to the set B. Symmetrical difference sets A and B is called the set A Δ B, which is the union of the differences of the sets AB and BA, that is, A Δ B = (AB) ∪ (BA). A ∪ B = B ∪ A (A ∪ B) ∪ C = A ∪ (B ∪ C) In order to compare any two sets A and B, a correspondence is established between their elements. If this correspondence is one-to-one, then the sets are called equivalent or equally powerful, A B or B A. The set of points on the leg BC and the hypotenuse AC of triangle ABC are of equal power.3. Set difference
4. Universal set
5. Set completion
6. Partitioning a set
7. Set algebra identities
Addition to the lesson “operations on sets”
Ordered set
Direct product of sets
Projection of a set.
The simplest cases
The coordinate line and number intervals as a union of their parts
for the intersection and union of numerical sets, respectively.
We have 
And 
The simplest cases
The coordinate line and number intervals as a union of their parts
And 
During the classes.
1. Organizational moment.
2. The teacher announces the topic of the lesson and formulates goals and objectives together with the students.
3. The teacher, together with the students, recalls the material studied on the topic “Sets” in 7th grade, introduces new concepts and definitions, formulas for solving problems.
Find the sets AU B,
A - (k, a, p, y, s, e, l, b).5. Summing up the lesson.
6. Reflection.
7. Homework.
If set A is part of set B, then write A ⊂ B (⊂
- contained).Basic number sets
N
(1,2,3,...,n) Set of all
Z
(0, ±1, ±2, ±3,...) Set integers. The set of integers includes the set of natural numbers.
Q
R
Elements of logical symbolism
Set Operations
For example, if A=(1,2,3,4), B=(3,1,4,2) then A=B.
For example, if A=(1,2,4), B=(3,4,5,6), then A ∪ B = (1,2,3,4,5,6)
For example, if A=(1,2,4), B=(3,4,5,2), then A ∩ B = (2,4)
For example, if A=(1,2,3,4), B=(3,4,5), then AB = (1,2)
For example, if A=(1,2,3,4), B=(3,4,5,6), then A Δ B = (1,2) ∪ (5,6) = (1,2,5 ,6)
Properties of set operations
Commutability properties
A ∩ B = B ∩ A
(A ∩ B) ∩ C = A ∩ (B ∩ C)Countable and uncountable sets