Question

If $A\cup B=A\cup C$ and $A\cap B=A\cap C$, then prove that $B=C$.

Answer 1

Let $x\in B......\left(1\right)$

$\Rightarrow x\in A\cup B$ [Since $B\subset A\cup B$, all elements of B are in $A\cup B$]

$\Rightarrow x\in A\cup C$ [Given $A\cup B=A\cup C]$

$\Rightarrow x\in A$ or $x\in C.....\left(2\right)$

Taking $x\in A$

$Also,x\in B$ [From (1)]

$\therefore x\in A\cap B$ [If x belongs to both A and B, it will belong to common of A and B also]

$x\in A\cap B$

So, $x\in A\cap C$ [Given $A\cup B=A\cup C$]

i.e. $x\in A$ and $x\in C$

i.e $x\in C$

$\therefore$ If $x\in B$, then $x\in C$

i.e. if an element belongs to set B, then it must belong to set C also

$\therefore B\subset C$

Taking $x\in C$

Also, $x\in B$ [From (1)]

$\therefore$ If $x\in B$, then $x\in C$

i.e if an elements belongs to set B, then it must belong to set C also

$\therefore B\subset C$

Similarly, we can prove $C\subset B$

Since $B\subset C$ and $C\subset B$

$\therefore B=C$

Hence proved.