Question

Let $A,B,$ and $C$ be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$. Show that $B=C$

Answer 1

Let, A, B and C be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$.
To show: $B=C$
Let $x\in B$
$\Rightarrow x\in A\cup B$ (By def of union of sets)
$\Rightarrow x\in A\cup C$ ($A\cup B=A\cup C$)
$\Rightarrow x\in A$ or $x\in C$
Case I
$x\in A$
Also, $x\in B$
$\therefore x\in A\cap B$
$\Rightarrow x\in A\cap C$ ($\because A\cap B=A\cap C$)
$\therefore x\in A$ and $x\in C$
$\therefore x\in C$
But $x$ is an arbitrary element in B.
$\therefore B\subset C$ .....(1)
Now, we will show that $C\subset B$.
Let $y\in C$
$\Rightarrow y\in A\cup C$ (by def of union of sets)
$\Rightarrow y\in A\cup B$ ($A\cup B=A\cup C$)
$\Rightarrow y\in A$ or $y\in B$
Case I : When $y\in A$
Also, $y\in C$
$\Rightarrow y\in A\cap C$
$\Rightarrow y\in A\cap B$
$\Rightarrow y\in A$ and $y\in B$
$\Rightarrow y\in B$
But $y$ is an arbitrary element of C.
Hence, $C\subset B$ .....(2)
From (1) and (2), w eget
$\therefore B=C$.