Intersection of Sets

The intersection of sets is known as a set with the common elements of both sets. It is represented by the symbol “∩”. For any two sets P and Q, the intersection, P ∩ Q, also read as P intersection Q, lists all the common elements of P and Q. For example, if a set P = {2, 4, 6, 8, 10, 12} and set Q = {3, 6, 9, 12}, then P ∩ Q = {6, 12}.

Example of the intersection of sets

1. i) If A = {a, e, i, o, u} and B = {a, b, c, d, e} then, A ∩ B= {a,e}
2. ii) If A = {the colors of the rainbow} = {Violet, Indigo, Blue, Green, Yellow, Orange, Red} and B = {Black, Red, Blue, White} then, A ∩ B = {Red, Blue}.

Disjoint sets

Consider two sets P and Q which have no common elements then these types of sets are known as disjoint sets. If P ∩ Q = ϕ, then P and Q are called disjoint sets. For example, If P = {2, 4, 6, 8} and Q = {1, 3, 5, 7, 9}, P ∩ Q = {2, 4, 6, 8} ∩ {1, 3, 5, 7, 9} = ϕ. Hence, P and Q are disjoint sets.

Cardinal number

The cardinal number of a set is described as the number of elements added up in a set. The cardinal number of a set X is represented as n(X). For example, if set X = {6, 7, 8, 9, 10}, then the cardinal number is represented as n (X) = 5. Consider two sets X and Y, X = {2, 4, 6, 8, 10, 12}, Y = {3, 6, 9, 12, 15} and X ∩ Y = {6, 12} where n (X ∩ Y) = 2. Thus, n (X ∩ Y) = n(X) + n(Y) – n (X ∪ Y).

Subsets

Consider, set P is the set of whole numbers from 0 to 20 and set Q is the set of even numbers from 0 to 20, then Q is the subset of P. The subset of every set constructing the intersection is an intersection of sets (P ∩ Q) ⊂ P and (P ∩ Q) ⊂ Q.

For example, P = {1, 2, 3, 4, 5, 6, 7, 8}, Q = {2, 4, 6, 8, 10}, P ∩ Q = {2, 4, 6, 8}. Therefore, P∩Q is a subset of P and Q.

Venn diagrams

Venn diagrams are diagrams that constitute or describe the connection between set operations. A Venn diagram is an extensively used diagram that shows the logical relation between sets. Circles represent each set of the Venn diagrams. Overlapping Venn diagrams or circles represent the typical elements present in two sets or more. Non-overlapping Venn diagrams or circles show no common elements between the two sets. 

Complement of the intersection of sets

The complement of the intersection of the given sets is the union of the sets except their intersection. The complement of intersection of sets A and B is denoted as (A∩B)’. For example, if A = {1, 2, 3, 4, 5}, B = {2, 4, 5, 6, 7, 8}, and A U B = {1, 2, 3, 4, 5, 6, 7, 8}, then A ∩ B= {2, 4, 5} and (A ∩ B)’ = {1, 3, 6, 7, 8}. 

Properties of the intersection of sets

Here is the list of properties of the intersection of sets, the name of property/law and its rule.

• Commutative Law: The union of two sets X and Y follows the commutative law, i.e., X ∩ Y = Y ∩ X
• Associative Law: The intersection of operations always follows the associative law, i.e., if we have three sets X, Y, and Z, the associative law will be defined as (X ∩ Y) ∩ C = X ∩ (Y ∩ C)
• Identity Law (or Law of ϕ and U): The identity laws state that just like 0 and 1 for addition and multiplication, ∅ and U are the identity elements for union and intersection. Union and intersection do not have inverse elements, i.e., ϕ ∩ X = ϕ, U ∩ X= X.
• Distribution Law: According to distribution law, if we have 3 sets, then (X ∩ Y) ∩ Z = (X ∩ Y) U (X – Z).
• Idempotent Law: The intersection of any set A with itself obtains the set A, i.e., A ∩ A = A. Like, (A ∩ (BUC) = (A ∩ B) ∪ (AUC), (A ∪ (B∩C) = (A ∪ B) ∩ (A∪C)
• Law of U: The union of universal set U with any set A obtains the set itself, i.e. X ∪  U = X.

Solved examples

Example 1: 

Let A = {1, 2, 3, 4, 6, 7, 9, 11}, B = {first five odd numbers}. Find A ∩ B and n (A ∩ B).

Solution:

Given A = {1, 2, 3, 4, 6, 7, 9, 11} and B = {first five odd numbers} = {1, 3, 5, 7, 9}. Therefore, A ∩ B = {1, 3, 7, 9} and thus, n (A ∩ B) = 4.

Example 2: 

If Set X = {1, 2, 3, 4, 5, 7, 9} and Set Y = {2, 4, 6, 8}. Find n (X ∩ Y).

Solution:

Given: Set X = {1, 2, 3, 4, 5, 7, 9} and Set Y = {2, 4, 6, 8}. Thus, X ∩ Y = {2, 4}. Then, n (X ∩ Y) = 2.

Therefore, n (X ∩ Y) = 2.

Example 3: 

If X = {1, 4, 6, 8, 10}, Y = {2, 4, 5, 7, 9}, and U = {1, 2, 4, 5, 6, 7, 8, 9, 10}. Find X ∩ Y and (X ∩ Y)’.

Solution:

Given: X = {1, 4, 6, 8, 10}, Y = {2, 4, 5, 7, 9}, and U= {1, 2, 4, 5, 6, 7, 8, 9, 10}. Then, X ∩ Y = {4}

⇒ (X ∩ Y)’ = {1, 2, 5, 6, 7, 8, 9, 10}.

 Conclusion

• In any two sets, X and Y, (X ∩ Y) is the set of elements common to both sets X and Y.
• If X ∩ Y = ϕ, then X and Y are disjoint sets.
• n (X ∩ Y) = n(X) + n(Y) – n (X ∪ Y)