Intersection of Sets

The symbol “∩” can be used to represent the intersection of two sets. The intersection of two sets A and B can be explained as the set of all elements that are common of both sets – A and B. A∩B is a symbol that represents the intersection of sets – A and B.

The intersection, A∩B (read as A intersection B) lists all the items that are present in both sets and are the common elements of A and B for any two sets A and B.

Example – Set A = {a, b, c, d, e} and Set B = {d, e, f, g}  

Hence, A∩ B = {d, e}

Intersection of Two Sets:

The most considerable set containing all the elements common to A and B is the intersection of two given sets, say A and B. The intersection of two sets can be a set with at least one element or an empty set with no items. If A and B are two sets with the property A ∩ B = φ, they are referred to as disjoint sets. That is, at the intersection of A and B, there are no elements.

Intersection of Three Sets:

Finding the intersection of more than two sets is achievable. You’ll learn how to find the intersection of three sets in this section. If A, B, and C are three sets, then the set of all elements that are common to A, B, and C is the intersection of these three sets. A ∩B ∩C can be used to symbolise this.

Properties of Intersection of Sets

The properties of the intersection of the sets are as follows: 

1. Commutative law
2. Associative law
3. Idempotent law
4. Law of φ and U
5. Distributive law

Let’s take a look at each of these properties one by one:-

1.Commutative law: P∩Q = Q∩P

Consider two sets P = {2, 4, 6, 8} and Q = {2, 3, 6, 9}.

Now, P∩Q = {2, 4, 6, 8} ∩ {2, 3, 6, 9} = {2, 6}

Q∩P= {2, 3, 6, 9} ∩ {2, 4, 6, 8} = {2, 6}

Hence, P∩Q = Q∩P.

2. Associative law: (P∩Q) ∩ R = P ∩ (Q∩R)

Let P = {2, 3, 4, 5}, Q = {4, 5, 6, 7}, and R = {6, 7, 8, 9}.

Now, P∩Q = {2, 3, 4, 5} ∩ {4, 5, 6, 7} = {4, 5}

(P∩Q) ∩ R = {4, 5} ∩ {6, 7, 8, 9} = { } = φ

Similarly, Q ∩ R = {4, 5, 6, 7} ∩ {6, 7, 8, 9} = {6, 7}

P ∩ (Q ∩ R) = {2, 3, 4, 5} ∩ {6, 7} = { } = φ

Hence, (P∩Q) ∩ R = P ∩ (Q ∩ R)

3. Idempotent law: P∩P = P

Suppose P = {w, x, y, z} such that P ∩ P = {w, x, y, z} ∩ {w, x, y, z} = {w, x, y, z} = P

4. 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

5. Distributive law: P ∩ (Q U R) = (P∩Q) U (P ∩ R)

Let us take three sets P = {1, 3, 6, 9}, Q = {2, 5, 7, 9} and R = {4, 5, 6, 9}.

Q U R = {2, 5, 7, 9} U {4, 5, 6, 9} = {2, 4, 5, 6, 7, 9}

P ∩ (Q U R) = {1, 3, 6, 9} ∩ {2, 4, 5, 6, 7, 9} = {6, 9}

And, P ∩ Q = {1, 3, 6, 9} ∩ {2, 5, 7, 9} = {9}

P ∩ R = {1, 3, 6, 9} ∩ {4, 5, 6, 9} = {6, 9}

(P∩Q) U (P ∩ R) = {9} U {6, 9} = {6, 9}

Hence, P ∩ (Q U R) = (P∩Q) U (P∩R)

Conclusion 

The most extensive set containing all the elements common to P and Q is the intersection of two given sets, say P and Q. The intersection of two sets is represented by the symbol “∩”.