Class 11

Math

Co-ordinate Geometry

Sets

Let $A,B,$ and $C$ be the sets such that $A∪B=A∪C$ and $A∩B=A∩C$. Show that $B=C$

To show: $B=C$

Let $x∈B$

$⇒x∈A∪B$ (By def of union of sets)

$⇒x∈A∪C$ ($A∪B=A∪C$)

$⇒x∈A$ or $x∈C$

Case I

$x∈A$

Also, $x∈B$

$∴x∈A∩B$

$⇒x∈A∩C$ ($∵A∩B=A∩C$)

$∴x∈A$ and $x∈C$

$∴x∈C$

But $x$ is an arbitrary element in B.

$∴B⊂C$ .....(1)

Now, we will show that $C⊂B$.

Let $y∈C$

$⇒y∈A∪C$ (by def of union of sets)

$⇒y∈A∪B$ ($A∪B=A∪C$)

$⇒y∈A$ or $y∈B$

Case I : When $y∈A$

Also, $y∈C$

$⇒y∈A∩C$

$⇒y∈A∩B$

$⇒y∈A$ and $y∈B$

$⇒y∈B$

But $y$ is an arbitrary element of C.

Hence, $C⊂B$ .....(2)

From (1) and (2), w eget

$∴B=C$.