Question

Let A,B and C be the sets such that A union B is equal to A union C and A intersection B is equal to A intersection C. show that B is equal to C

Answer 1

Lets Set A contain n elements
$\therefore$ Set A : n A
Similarly, Set B : n B
Set C : n C
Let, uas assume that, $SetB\ne SetC\Rightarrow nB\ne nC$
$\Rightarrow nB>nC$ or nB < nC
Let, nB > nC
Now, $A\cup B=nA+nB-n(A\cap B)$
$A\cup =nA+nC-n(A\cap C)$
But, $(A\cap B)=(A\cap C)$ (given)
$\therefore (A\cup B)>(A\cup C)$ [As nB > nC]
But it is given given that, $(A\cup B)=(A\cup C)$
Which contradicts our hypotheris that $nB\ne nC$
$\therefore nB=nC$
$\Rightarrow$ Set B = Set C