Intersection of Sets
The intersection of two given sets is the set that contains all the elements that are common to both sets. The symbol for the intersection of sets is "∩''. For any two sets A and B, the intersection, A ∩ B (read as A intersection B) lists all the elements that are present in both sets (common elements of A and B).
For example, if Set A = {1,2,3,4,5} and Set B = {3,4,6,8}, A ∩ B = {3,4}. Let us earn more about the properties of the intersection of sets along with examples.
| 1. | What is Intersection of Sets? |
| 2. | Complement of Intersection of Sets |
| 3. | Intersection of Sets Venn Diagram |
| 4. | Properties of Intersection of Sets |
| 5. | FAQs on Intersection of Sets |
What is Intersection of Sets?
The intersection of sets is the set of elements which are common to the given sets. In set theory, for any two sets A and B, the intersection is defined as the set of all the elements in set A that are also present in set B. We use the symbol '∩' that denotes 'intersection of'. For example, let us represent the students who like ice creams for dessert, Brandon, Sophie, Luke, and Jess. This is set A. The students who like brownies for dessert are Ron, Sophie, Mia, and Luke. This is set B. The students who like both ice creams and brownies are Sophie and Luke. This is represented as A ∩ B.
Disjoint Sets
Two sets A and B having no elements in common are said to be disjoint, if A ∩ B = ϕ, then A and B are called disjoint sets. Example: If A = { 2, 3, 5, 9} and B = {1, 4, 6,12}, A ∩ B = { 2, 3, 5, 9} ∩ {1, 4, 6,12} = ϕ. Therefore, A and B are called disjoint sets.
Subsets
If set A is the set of natural numbers from 1 to 10 and set B is the set of odd numbers from 1 to 10, then B is the subset of A. The intersection of sets is a subset of each set forming the intersection, (A ∩ B) ⊂ A and (A ∩ B) ⊂ B.
For example- A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} , B = {2, 4, 7, 12, 14} , A ∩ B = {2, 4, 7}. Thus, A ∩ B is a subset of A, and A ∩ B is a subset of B.
A Intersection B Formula
The A intersection B formula talks about the cardinality of a set. The cardinal number of a set is the total number of elements present in the set. For example, if Set A = {1,2,3,4}, then the cardinal number (represented as n (A)) = 4. Consider two sets A and B. A = {2, 4, 5, 6,10,11,14, 21}, B = {1, 2, 3, 5, 7, 8,11,12,13} and A ∩ B = {2, 5, 11}, and the cardinal number of A intersection B is represented by n(A ∩ B) = 3.
The cardinality of A ∩ B can also be found by A intersection B formula which states: n(A ∩ B)= n(A) + n(B) - n(A ∪ B). We will verify this formula for the above example, where n(A) = 8, n(B) = 9, and A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 21}. Note that n(A ∪ B) = 14 here. Then
n(A ∩ B)= n(A) + n(B) - n(A ∪ B)
3 = 8 + 9 - 14
3 = 3
Hence, n(A intersection B) formula is verified. We can observe the same in the figure below.
Complement of Intersection of Sets
The set of all the elements in the universal set but not in A ∩ B is the complement of the intersection of sets. It is denoted by (A ∩ B)' and is read as A intersection B complement. The shaded region in blue in the following Venn diagram represents (A ∩ B)'.
Example: If X = {1, 2, 3, 4, 5}, Y = {2, 4, 6, 8, 10}, and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then X ∩ Y = {2,4} and (X ∩ Y)' = {1, 3, 5, 6, 7, 8, 9, 10}.
Intersection of Sets Venn Diagram
Intersection of sets can be easily understood using venn diagrams. Venn diagrams use circles to represent each set. Overlapping circles denote that there is some relationship between two or more sets, and that they have common elements. The circles that do not overlap do not share any common elements. The following diagram shows the intersection of sets using a Venn diagram. Here, Set A = {1, 2, 3, 4, 5} and Set B = {3, 4, 6, 8}. Therefore A ∩ B = {3 ,4}
Note: In the above figure, the portion in pink represents A ∩ B.
Properties of Intersection of Sets
Intersection of sets have properties similar to the properties of numbers. The properties of intersection of sets include the commutative law, associative law, law of null set (ϕ), universal set (U), and the idempotent law. The following table lists the properties (laws) of the intersection of sets.
| Law | Rule of Intersection |
|---|---|
| Commutative Law | A ∩ B = B ∩ A |
| Associative Law | (A ∩ B) ∩ C = A ∩ (B ∩ C) |
| Law of ϕ and U | ϕ ∩ A = ϕ, U ∩ A= A |
| Idempotent Law | A ∩ A = A |
| De Morgan's Law | (A ∩ B)' = A' U B' |
Important Notes:
- (A ∩ B) is the set of all the elements that are common to both sets A and B.
- If A ∩ B = ϕ, then A and B are called disjoint sets.
- n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
☛ Related Topics:
FAQs on Intersection of Sets
What is A Intersection B?
A intersection B represents the intersection of sets A and B is represented as A ∩ B. It is defined as the group of elements present in set A that are also present in set B. In the Venn diagram of A and B, A intersection B is the portion that is the common portion of both A and B.
What Does A ∩ B Mean in Math?
A ∩ B means the common elements that belong to both set A and set B. In math, ∩ is the symbol to denote the intersection of sets.
What is Union and Intersection of Sets?
For any two sets A and B, the union of sets, which is denoted by A U B, is the set of all the elements present in set A and the set of elements present in set B or both. The intersection of two sets is the set of elements that are common to both set A and set B.
What Does ∩ Mean in Probability?
If there are two events A and B, then ∩ denotes the probability of the intersection of the events A and B. In probability, P(A ∩ B) = P(A) · P(B) only when the events are independent.
What is A intersection A?
A intersection A is A itself. This is written as A ∩ A = A. This is called the idempotent law of the intersection of sets.
What Is the Formula of Intersection of Two Sets?
The intersection of sets formula gives the total number of elements in a set (which is called the cardinal number of the set). It says, for the two finite sets A and B, n(A ∩ B) = n(A) + n(B) – n(A ∪ B).
Is A ∩ B Equal to B ∩ A?
As per the commutative property of the intersection of sets, the order of the operating sets does not affect the resultant set and thus A ∩ B equals B ∩ A. For example, for the sets P = {a, b, c, d, e},and Q = {a, e, i}, A ∩ B = {a,e} and B ∩ A = {a.e}. Thus, A ∩ B = B ∩ A.
What are the Formulas Associated With Intersection of 3 Sets?
The intersection of 3 sets (A intersection B intersection C) is associative. It means it can be computed in any order. i.e., (A ∩ B) ∩ C = A ∩ (B ∩ C). The following two are common formulas associated with 3 sets that include both union and intersection.
- A intersection B union C: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- A union B Intersection C: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
What Is the Symbol of Intersection of Sets?
The mathematical symbol that is used to represent the intersection of two sets is ' ∩'.
What Is the Complement of Intersection of Sets?
The complement of set A ∩ B is the set of elements that are members of the universal set U but not members of set A ∩ B. In other words, the complement of the intersection of the given sets is the union of the sets excluding their intersection. It is represented as (A∩B)´.