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:
- Commutative law
- Associative law
- Idempotent law
- Law of φ and U
- 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 “∩”.