Some solution mathematical problems involves finding intersection and union number 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.

The simplest cases

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 original sets.

From these 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

1. Let us define the union of the original sets. Let's write down all the elements, for example, of set A: 3, 5, 7, 12. Let's add to them the missing elements of set B: 2, 8, 11 and 13. Ultimately, we have a numerical set: (3, 5, 7, 12, 2, 8, 11, 13). Let's order the elements of the resulting set and get the desired union: A ∪ B = (2, 3, 5, 7, 8, 11, 12, 13).
2. Let us define the intersection of the original sets. According to the rule, we will go through all the elements of the first set A one by one and check whether they are included in the set B. Let's consider the first element - the number 3: it does not belong to the set B, which means it will not be an element of the desired intersection. Let's check the second element of set A, i.e. number 5: it belongs to the set B, which means it will become the first element of the desired intersection. The third element of set A is the number 7. It is not an element of the set B, and, therefore, is not an element of intersection. Consider the last element of set A: the number 1. It also belongs to the set B, and accordingly will become one of the intersection elements. Thus, the intersection of the original sets is a set consisting of two elements: 5 and 12, i.e. A ∩ B = (5, 12).

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 ) .

The coordinate line and number intervals as a union of their parts

Let's mark an arbitrary point on the coordinate line, for example, with coordinates - 5, 4. 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.

Note that it is possible to attach a given point to any of the open rays, 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 will look at a general approach that allows us to determine the result of the intersection and union of two number 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

1. Let us depict the given numerical sets on coordinate lines. They need to be placed one above the other. For convenience, it is generally accepted that the origin points of the given sets coincide, and the location of the points relative to each other remains preserved: any point with a larger coordinate lies to the right of the point with a smaller coordinate. Moreover, if we are interested in the union of sets, then the coordinate lines are combined on the left by the square bracket of the set; if you are interested in intersection, then use the curly brace of the system.

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.

And

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.

1. Let's go back to the example and define the intersection of given sets. To do this, let's check the sets one by one: (- ∞ , - 3) , ( - 3 ) , (- 3 , 7) , ( 7 ) , (7 , + ∞) . Let's start with the set (- ∞, - 3), clearly highlighting it in the drawing:

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 more number 7, i.e.: A ∩ B = (7, + ∞).

1. Next step Let's define the union of the given sets A and B. We will sequentially check the sets (- ∞ , - 3), ( - 3), (- 3, 7), ( 7), (7, + ∞), establishing the fact of their inclusion or non-inclusion in the desired union.

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 integral part set B (hatching is present 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 the final result. Let us demonstrate with a practical example what its 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

1 QUESTION:Many is a collection of certain elements united by some common feature. Elements of a set can be numbers, figures, objects, concepts, etc.

Sets are denoted by uppercase letters, and elements of the set are denoted by lowercase letters. Elements of sets are enclosed in curly braces.

If element x belongs to the set X, then write x ∈ X (∈ - belongs). If set A is part of set B, then write A⊂ IN (⊂ - contained).

Definition 1 (definition of equality of sets). Sets A and B are equal if they consist of the same elements, that is, if x  A implies x  B and vice versa, x  B implies x  A.

Formally, the equality of two sets is written as follows:

(A=B):= x((xA)  (xB)),

this means that for any object x the relations x A and x B are equivalent.

Here  is the universal quantifier ( x reads "for everyone" x").

Subset

Definition: The set X is subset Y, if any element of the set X belongs to the set Y. This is also called non-strict inclusion.Some properties of the subset:

1. ХХ - reflectivity

2. X  Y & YZ  X  Z - transitivity

3.   X i.e. the empty set is a subset of any set. Universal set Definition: Universal set- this is a set that consists of all elements, as well as subsets of the set of objects in the area under study, i.e.

1. If M  I , That M  I

2. If M  I , That Ώ(M)  I, where under Ώ(M) - all possible subsets of M, or Boolean M, are understood.

The universal set is usually denoted I .

The universal set can be chosen independently, depending on the set under consideration and the tasks being solved.

Methods for specifying sets:

1. by listing its elements. Usually finite sets are defined by enumeration.

2. by describing properties common to all elements of this set, and only this set. This property is called characteristic property, and this way of specifying the set description. Thus, you can specify both finite and infinite sets. If we define a set with some property, then it may turn out that only one object has this property or that there is no such object at all. This fact may not be at all obvious.

Topic 2.3 Operations on sets.

Now let's define operations on sets.

1. Intersection of sets.

Definition: The intersection of the sets X and Y is a set consisting of all those, and only those elements, that belong to both the set X and the set Y.

For example: X=(1,2,3,4) Y=(2,4,6) intersection (2,4)

Definition: Sets are called disjoint if they do not have common elements, i.e. their intersection is equal to the empty set.

For example : disjoint sets are the sets of excellent students and unsuccessful ones.

This operation can be extended to more than two sets. In this case, it will be a set of elements that simultaneously belong to all sets.

Intersection properties:

1. X∩Y = Y∩X - commutativity

2. (X∩Y) ∩Z =X∩ (Y∩Z)=X∩Y∩Z - associativity

3. X∩ = 

4. X∩ I = X

2. Union of sets

Definition: The union of two sets is a set consisting of all and only those elements that belong to at least one of the sets X or Y.

For example: X=(1,2,3,4) Y=(2,4,6) by combining (1,2,3,4,6)

This operation can be extended to more than two sets. In this case, it will be the set of elements belonging to at least one of these sets.

Join properties:

1. XUY= YUY - commutativity

2. (X UY)UZ =XU (YUZ)=XUYUZ - associativity

4.XU I = I

From the properties of the operations of intersection and union it is clear that the empty set is similar to zero in the algebra of numbers.

3. Set difference

Definition: This operation, unlike the operations of intersection and union, is defined only for two sets. The difference between the 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.

For example: X=(1,2,3,4) Y=(2,4,6) difference (1,3)

As we have already seen, the role of zero in set algebra is played by the empty set. Let us define a set that will play the role of unit in the algebra of sets

4. Set completion

The complement of a set X is the difference between I and X.

Add-on properties:

1. The set X and its complement have no common elements

2. Any element I belongs either to the set X or its complement.

QUESTION 2 Sets of numbers

Integers− numbers used when counting (listing) items: N=(1,2,3,…)

Natural numbers with zero included− numbers used to indicate the number of items: N0=(0,1,2,3,…)

Whole numbers− include integers, numbers opposite to natural ones (i.e. with a negative sign) and zero. Positive integers: Z+=N=(1,2,3,…) Negative integers: Z−=(…,−3,−2,−1) Z=Z−∪(0)∪Z+=(…,−3,−2,−1,0,1,2,3,…)

Rational numbers− numbers represented as a common fraction a/b, where a and b are integers and b≠0. Q=(x∣x=a/b,a∈Z,b∈Z,b≠0) When converted to decimal a rational number is represented by a finite or infinite periodic fraction.

Irrational numbers− numbers that are represented as an infinite non-periodic decimal fraction.

Real numbers− union of rational and irrational numbers: R

Complex numbers C=(x+iy∣x∈R иy∈R), where i is the imaginary unit.

Real number modulus and properties

Modulus of a real number- This absolute value this number.

Simply put, when taking the modulus, you need to remove its sign from the number.

The absolute value of a number a denoted by |a|. Please note: the modulus of a number is always non-negative: |a|≥ 0.

|6| = 6, |-3| = 3, |-10,45| = 10,45

The concept of set theory; intersection of sets is a set consisting of all those elements that simultaneously belong to all given sets. The intersection of sets A and B is denoted by A?B or AB...

The concept of set theory; intersection of sets is a set consisting of all those elements that simultaneously belong to all given sets. The intersection of sets A and B is denoted by A∩B or AB. * * * INTERSECTION OF SET INTERSECTION OF SET… encyclopedic Dictionary

A set consisting of all those elements that simultaneously belong to all given sets. P. m. A and B denote A∩B or AB; P. m. Ak, taken in a finite or infinite number, are denoted by Ak. P.m. can be empty, that is, not... ... Great Soviet Encyclopedia

The concept of set theory; P. m. is a set consisting of all those elements to which they belong at the same time. to all given sets. P.m...

The intersection of A and B The intersection of sets in set theory is a set consisting of elements that simultaneously belong to all given sets. Contents 1 Definition 2 Note ... Wikipedia

Branch of mathematics in which they study general properties sets, mostly infinite. the concept of a set is the simplest mathematical concept, it is not defined, but only explained with the help of examples: a lot of books on a shelf, a lot of points... Big Encyclopedic Dictionary

A branch of mathematics that studies the general properties of sets, especially infinite ones. The concept of a set is the simplest mathematical concept; it is not defined, but only explained with the help of examples: many books on a shelf, many... ... encyclopedic Dictionary

A mathematical theory that studies the problem of infinity by precise means. Subject M. l. properties of sets (collections, classes, ensembles), ch. arr. endless. A set A is any collection of defined and distinguishable objects... Dictionary of Logic Terms

Set theory is a branch of mathematics that studies the general properties of sets. Set theory underlies most mathematical disciplines; it had a profound influence on the understanding of the subject of mathematics itself. Contents 1 Theory ... ... Wikipedia

A branch of mathematics in which the general properties of sets are studied, especially. endless. The concept of set is the simplest math. concept, it is not defined, but only explained with the help of examples: many books on a shelf, many points on a straight line... ... Natural science. encyclopedic Dictionary

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