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 $B = C$

Vedclass pdf generator app on play store
Vedclass iOS app on app store

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 $x \in B$

$\Rightarrow x \in A \cup B \quad[B \subset A \cup B]$

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

$\Rightarrow x \in A$ or $x \in C$

Case $I$

Also, $x \in B$

$\therefore x \in A \cap B$

$\Rightarrow x \in A \cap C \quad[\because A \cap B=A \cap C]$

$\therefore x \in A$ and $x \in C$

$\therefore x \in C$

$\therefore B \subset C$

Similarly, we can show that $C \subset B$

$\therefore B=C$

Similar Questions

Examine whether the following statements are true or false :

$\{ 1,2,3\}  \subset \{ 1,3,5\} $

Let $A=\{1,2,3,4,5,6\} .$ Insert the appropriate symbol $\in$ or $\notin$ in the blank spaces:

$ 8\, .......\, A $

Consider the sets

$\phi, A=\{1,3\}, B=\{1,5,9\}, C=\{1,3,5,7,9\}$

Insert the symbol $\subset$ or $ \not\subset $ between each of the following pair of sets:

$A, \ldots B$

Are the following pair of sets equal ? Give reasons.

$A = \{ 2,3\} ,\quad \,\,\,B = \{ x:x$ is solution of ${x^2} + 5x + 6 = 0\} $

Examine whether the following statements are true or false :

$\{ a\}  \in \{ a,b,c\} $