Intersection of Set, Properties of Intersection - Practice Questions & MCQ

The intersection of sets $A$ and $B$ is the set of all elements which are common to both $A$ and $B$. The symbol ' $\cap$ 'is used to denote the intersection.

Symbolically, we write $A\cap B=\{x:x\in A$ and $x\in B\}$
For example, let $\mathrm{A}=\{2,4,6,8\}$ and $\mathrm{B}=\{2,3,5,8\}$, then $\mathrm{A}\cap \mathrm{B}=\{2,8\}$

If $A$ and $B$ are two sets such that $A\cap B=\phi$, then $A$ and $B$ are called disjoint sets.
For example, let $A=\{2,4,6,8\}$ and $B=\{1,3,5,7\}$. Then $A$ and $B$ are disjoint sets because there are no elements which are common to A and B .

Properties of intersection
$\mathrm{A}\cap \mathrm{B}=\mathrm{B}\cap \mathrm{A}$ (Commutative law).
$(A\cap B)\cap C=A\cap (B\cap C)$ (Associative law).
$\mathrm{A}\cap \varphi =\varphi$,
$\mathrm{A}\cap \mathrm{U}=\mathrm{A}$ (Law of $\varphi$ and U$)$.
$\mathrm{A}\cap \mathrm{A}=\mathrm{A}$ (Idempotent law)
If $A$ is subset of $B$, then $A\cap B=A$

Distributive laws

1. $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$ i. e., $\cap$ distributes over $\cup$

This can be seen easily from the following Venn diagrams

LHS:

RHS:

2. $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

This can be seen easily from the following Venn diagrams

LHS:

RHS: