The set operations are performed on two or more sets to obtain a combination of elements as per the operation performed on them. In a set theory, there are three major types of operations performed on sets, such as:
- Union of sets (∪)
- Intersection of sets (∩)
- Difference of sets ( – )
Let us discuss these operations one by one.
Union of Sets
If two sets A and B are given, then the union of A and B is equal to the set that contains all the elements present in set A and set B. This operation can be represented as;
A ∪ B = {x: x ∈ A or x ∈ B}
Where x is the elements present in both sets A and B.
Example: If set A = {1,2,3,4} and B {6,7}
Then, Union of sets, A ∪ B = {1,2,3,4,6,7}
Venn Diagram of Union of sets
Intersection of Sets
If two sets A and B are given, then the intersection of A and B is the subset of universal set U, which consist of elements common to both A and B. It is denoted by the symbol ‘∩’. This operation is represented by:
A∩B = {x : x ∈ A and x ∈ B}
Where x is the common element of both sets A and B.
The intersection of sets A and B, can also be interpreted as:
A∩B = n(A) + n(B) – n(A∪B)
Where,
n(A) = cardinal number of set A,
n(B) = cardinal number of set B,
n(A∪B) = cardinal number of union of set A and B.
Example: Let A = {1,2,3} and B = {3,4,5}
Then, A∩B = {3}; because 3 is common to both the sets.
Venn Diagram of Intersection of sets
Intersection of Two Sets
Intersection of three sets
Difference of Sets
If there are two sets A and B, then the difference of two sets A and B is equal to the set which consists of elements present in A but not in B. It is represented by A-B.
Example: If A = {1,2,3,4,5,6,7} and B = {6,7} are two sets.
Then, the difference of set A and set B is given by;
A – B = {1,2,3,4,5}
We can also say, that the difference of set A and set B is equal to the intersection of set A with the complement of set B. Hence,
A−B=A∩B’
Venn Diagram of Difference of sets
Also, read:
Complement of Set
If U is a universal set and X is any subset of U then the complement of X is the set of all elements of the set U apart from the elements of X.
X′ = {a : a ∈ U and a ∉ A}
Venn Diagram:
Example: U = {1,2,3,4,5,6,7,8}
A = {1,2,5,6}
Then, complement of A will be;
A’ = {3,4,7,8}
Video Lesson on What are Sets
Properties of Set Operations
Commutative Property
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Associative Property
A ∪ (B ∪ C) = (A ∪ B) ∪ C
A ∩ (B ∩ C) = (A ∩ B) ∩ C
Distributive Property
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
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