{ , }. Example Consider the sets X and Y of Example . Find X ∩ Y. Solution We see that element ‘Geeta’ is the only element common to both.
Hence, X ∩ Y = {Geeta}. Example Let A = { , , , , , , , , , } and B = { , , , }. Find A ∩ B and hence show that A ∩ B = B. Solution We have A ∩ B = { , , , } = B.
We note that B ⊂ A and that A ∩ B = B. Definition The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write A ∩ B = { x : x ∈ A and x ∈ B} The shaded portion in Fig . indicates the intersection of A and B.
Fig . Fig . MATHEMATICS If A and B are two sets such that A ∩ B = φ , then A and B are called disjoint sets. For example, let A = { , , , } and B = { , , , }.
Then A and B are disjoint sets, because there are no elements which are common to A and B. The disjoint sets can be represented by means of Venn diagram as shown in the Fig . In the above diagram, A and B are disjoint sets. Some Properties of Operation of Intersection (i) A ∩ B = B ∩ A (Commutative law).
(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law). (iii) φ ∩ A = φ , U ∩ A = A (Law of φ and U). (iv) A ∩ A = A (Idempotent law) (v) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e., ∩ distributes over ∪ This can be seen easily from the following Venn diagrams