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

We know that $A=A\cap (A\cup B)$ and $A=A\cup (A\cap B)$

Now $A\cap B=A\cap C$ and $A\cup B=A\cup C$

$\therefore B=B\cup (B\cap A)=B\cup (A\cap B)$

$=B\cup (A\cap C)[\because A\cap B=A\cap C]$

$=(B\cup A)\cap (B\cup C)$ (By distributive law)

$=(A\cup B)\cap (B\cup C)$

$=(A\cup C)\cap (B\cup C)$ $[\because A\cup B=A\cup C]$

$=(C\cup A)\cap (C\cup B)$

$=C\cup (A\cap B)$ (By distributive law)

$=C\cup (A\cap C)$ $[\because A\cap B=A\cap C]$

$=C\cup (C\cap A)=C$ Hence $B=C$