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Intersection of Set, Properties of Intersection - (Concept)

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 $A \cap B=\varphi$, 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).
$\mathrm{A} \cap \phi=\phi$,
$\mathrm{A} \cap \mathrm{U}=\mathrm{A}$ (Law of $\phi$ 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:

           

 

 

 

 

Exam Chapter
JEE MAIN Sets, Relations and Functions
Mathematics Textbook for Class XI
Page No. : 15
Line : 15

Algebra (Arihant)
Page No. : 789
Line : 4

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