When dealing with set theory, there are a number of operations to make new sets out of old ones. One of the most common set operations is called the intersection. Simply stated, the intersection of two sets A and B is the set of all elements that both A and B have in common. Show
We will look at details concerning the intersection in set theory. As we will see, the key word here is the word "and." For an example of how the intersection of two sets forms a new set, let’s consider the sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 8}. To find the intersection of these two sets, we need to find out what elements they have in common. The numbers 3, 4, 5 are elements of both sets, therefore the intersections of A and B is {3. 4. 5]. In addition to understanding the concepts concerning set theory operations, it is important to be able to read symbols used to denote these operations. The symbol for intersection is sometimes replaced by the word “and” between two sets. This word suggests the more compact notation for an intersection that is typically used. The symbol used for the intersection of the two sets A and B is given by A ∩ B. One way to remember that this symbol ∩ refers to intersection is to notice its resemblance to a capital A, which is short for the word "and." To see this notation in action, refer back the above example. Here we had the sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 8}. So we would write the set equation A ∩ B = {3, 4, 5}. One basic identity that involves the intersection shows us what happens when we take the intersection of any set with the empty set, denoted by #8709. The empty set is the set with no elements. If there are no elements in at least one of the sets we are trying to find the intersection of, then the two sets have no elements in common. In other words, the intersection of any set with the empty set will give us the empty set. This identity becomes even more compact with the use of our notation. We have the identity: A ∩ ∅ = ∅. For the other extreme, what happens when we examine the intersection of a set with the universal set? Similar to how the word universe is used in astronomy to mean everything, the universal set contains every element. It follows that every element of our set is also an element of the universal set. Thus the intersection of any set with the universal set is the set that we started with. Again our notation comes to the rescue to express this identity more succinctly. For any set A and the universal set U, A ∩ U = A. There are many more set equations that involve the use of the intersection operation. Of course, it's always good to practice using the language of set theory. For all sets A, and B and D we have:
This lesson will explain how to find the intersection of sets. We will start with a definition of the intersection of two sets. Definition: Given two sets A and B, the intersection is the set that contains elements or objects that belong to A and to B at the same time. We write A ∩ B Basically, we find A ∩ B by looking for all the elements A and B have in common. Next, we illustrate with examples. Example #1. To make it easy, notice that what they have in common is in bold. Let A = { 1 orange, 1 pineapple, 1 banana, 1 apple } and B = { 1 spoon, 1 orange, 1 knife, 1 fork, 1 apple } A ∩ B = { 1 orange, 1 apple } Example #2. Find the intersection of A and B and then make a Venn diagrams. A = { b, 1, 2, 4, 6 } and B = { 4, a, b, c, d, f } A ∩ B = { 4, b } 
A = { x / x is a number bigger than 4 and smaller than 8 } B = { x / x is a positive number smaller than 7 } A = { 5, 6, 7 } and B = { 1, 2, 3, 4, 5, 6 } A ∩ B = { 5, 6 } Or A ∩ B = { x / x is a number bigger than 4 and smaller than 7 } Example #4. A = { x / x is a country in Asia } B = { x / x is a country in Africa } Since no countries in Asia and Africa are the same, the intersection is empty. A ∩ B = { } Example #5. A = { #, %, &, *, $ } B = { } This example is subtle! Since the empty set is included in any set, it is also included in A although you don't see it. Therefore, the empty set is the only thing set A and set B have in common.A ∩ B = { } Definition of the union of three sets: Given three sets A, B, and C the intersection is the set that contains elements or objects that belong to A, B, and to C at the same time.We write A ∩ B ∩ C Basically, we find A ∩ B ∩ C by looking for all the elements A, B, and C have in common. A = { #, 1, 2, 4, 6 }, B = { #, a, b, 4, c } and C = A = { #, %, &, *, $, 4 }A ∩ B ∩ C = { 4 , # } The graph below shows the shaded region for the intersection of two sets
In Mathematics, a set is a well-defined collection of objects. Unlike numbers, we can define different operations on sets and examine their properties. In set theory, an operation is an exercise to combine different sets such that a new set will be obtained with distinct properties. In this article, you will learn the intersection of sets definition, formula, expressions, along with properties and examples in detail. What is Intersection of Sets?The intersection of sets A and B is the set of all elements which are common to both A and B. Suppose A is the set of even numbers less than 10 and B is the set of the first five multiples of 4, then the intersection of these two can be identified as given below: A = {2, 4, 6, 8} B = {4, 8, 12, 16, 20} The elements common to A and B are 4 and 8. Therefore, the set of elements in the intersection A and B = {4, 8}
Intersection of Sets SymbolThe intersection of sets can be denoted using the symbol ‘∩’. As defined above, the intersection of two sets A and B is the set of all those elements which are common to both A and B. Symbolically, we can represent the intersection of A and B as A ∩ B. A intersection B FormulaThe formula for the intersection of sets can be given based on the definition. That means, the intersection of two sets A and B is the set containing elements that are common to both A and B. This can be expressed as: A ∩ B = {x : x ∈ A and x ∈ B} That means x is an element of A ∩ B, if and only if x is an element of both A and B. Thus, we can use the word “AND” to represent the intersection of sets. Sometimes, the above expression can also be referred to as the intersection of sets formula. Intersection of Two SetsThe intersection of two given sets, say X and Y, is the most extensive set containing all the elements common to X and Y. The intersection of two sets can be a set again with at least one element or it can be an empty set, i.e. no elements in the intersection set. Suppose A and B are two sets such that A ∩ B = φ, then A and B are called disjoint sets. That means, there are no elements at the intersection of A and B. Intersection of Three SetsIt is possible to find the intersection of more than two sets. In this section, you will learn how to find the intersection of three sets. Suppose A, B and C are three sets, then the intersection of these three sets is the set of all elements that are common to A, B and C. This can be represented as A ∩ B ∩ C. This can be better understood with the help of the example given below. For example, A = {6, 8, 10, 12, 14, 16}, B = {9, 12, 15, 18, 21, 24} and C = {4, 8, 12, 16, 20, 24, 28}. The intersection of these 3 sets can be expressed as, A ∩ B ∩ C. Let us identify the common elements of the given sets. 12 is the only common element in A, B and C. Therefore, A ∩ B ∩ C = {12} Intersection of Sets Using Venn DiagramThe intersection of two sets can be represented using the Venn diagram as shown below. In the above diagram, the shaded portion represents the intersection of two sets A and B. Similarly, we can draw a Venn diagram for the intersection of 3 sets as shown below. In the above diagram, we can see that the centermost region denotes the intersection of three sets A, B and C. Intersection of Sets PropertiesSome Properties of the Operation of Intersection are listed below: (i) Commutative law: A ∩ B = B ∩ A Consider two sets A = {1, 2, 3, 4, 5, 6} and B = {2, 3, 5, 7}. Now, A ∩ B = {1, 2, 3, 4, 5, 6} ∩ {2, 3, 5, 7} = {2, 3, 5} B ∩ A = {2, 3, 5, 7} ∩ {1, 2, 3, 4, 5, 6} = {2, 3, 5} Therefore, A ∩ B = B ∩ A. (ii) Associative law: (A ∩ B) ∩ C = A ∩ (B ∩ C) Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {5, 6, 7, 8 }. Now, A ∩ B = {1, 2, 3, 4} ∩ {3, 4, 5, 6} = {3, 4} (A ∩ B) ∩ C = {3, 4} ∩ {5, 6, 7, 8} = { } = φ And B ∩ C = {3, 4, 5, 6} ∩ {5, 6, 7, 8} = {5, 6} A ∩ (B ∩ C) = {1, 2, 3, 4} ∩ {5, 6} = { } = φ Therefore, (A ∩ B) ∩ C = A ∩ (B ∩ C) (iii) Law of φ and U: φ ∩ A = φ, U ∩ A = A Consider φ = { } and A = {10, 11, 12}. φ ∩ A = { } ∩ {10, 11, 12} = { } = φ Let U = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} and A = {4, 8, 12, 16, 20}. U ∩ A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} ∩ {4, 8, 12, 16, 20} = {4, 8, 12, 16, 20} = A (iv) Idempotent law: A ∩ A = A Suppose A = {a, b, c, d, e} such that A ∩ A = {a, b, c, d, e} ∩ {a, b, c, d, e} = {a, b, c, d, e} = A (v) Distributive law: A ∩ (B U C) = (A ∩ B) U (A ∩ C), i. e., ∩ distributes over U Let us take three sets A = = {2, 4, 6, 8}, B = {2, 3, 5, 7} and C = {3, 4, 5, 6}. B U C = {2, 3, 5, 7} U {3, 4, 5, 6} = {2, 3, 4, 5, 6, 7} A ∩ (B U C) = {2, 4, 6, 8} ∩ {2, 3, 4, 5, 6, 7} = {2, 4, 6} A ∩ B = {2, 4, 6, 8} ∩ {2, 3, 5, 7} = {2} A ∩ C = {2, 4, 6, 8} ∩ {3, 4, 5, 6} = {4, 6} (A ∩ B) U (A ∩ C) = {2} U {4, 6} = {2, 4, 6} Therefore, A ∩ (B U C) = (A ∩ B) U (A ∩ C) Intersection of Sets ExamplesGo through the examples given below to get a thorough understanding of the concept. Example 1: If A = { 3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, and C = {11, 13, 15}, then find B ∩ C and A ∩ B ∩ C. Solution: Given, A = { 3, 5, 7, 9, 11} B = {7, 9, 11, 13} C = {11, 13, 15} B ∩ C = {11, 13} A ∩ B ∩ C = {11} Example 2: Represent the following using Venn diagrams. (i) A ∩ (B ∪ C) (ii) (A ∩ B) ∪ (A ∩ C) Solution: Venn diagram of A ∩ (B ∪ C)
Venn diagram of (A ∩ B) ∪ (A ∩ C) Union and Intersection of SetsThe union of two sets A and B is the set of all those elements which are either in A or in B, whereas the intersection of two sets A and B is the set of all elements which are common. We can define the formulas for both union and intersection of given sets based on the cardinality of sets as given below. If A and B are finite sets such that A ∩ B = φ, then n (A ∪ B) = n (A) + n (B). If A ∩ B ≠ φ, then n (A ∪ B) = n (A) + n (B) – n (A ∩ B) Practice Problems
Video Lesson on What are SetsTo get more information about operations on sets, visit www.byjus.com and download BYJU’S – The Learning App to get interactive videos on Maths concepts.
The intersection operation on sets gives the set of elements that are common in the given sets.
The symbol ∩ denotes the intersection of sets. For example, the intersection of two sets X and Y can be represented as X ∩ Y.
The union of two sets A and B is the set of all those elements which are either in A or in B, i.e. A ∪ B, whereas the intersection of two sets A and B is the set of all elements which are common. The intersection of these two sets is denoted by A ∩ B.
The other word for intersection is AND which means we can represent the intersection of A, B as A and B or A ∩ B.
Yes, A ∩ B is equal to B ∩ A. This represents the commutative law of the intersection of sets. |
How do you write the intersection of two sets?
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