Let A, B, and C be the sets such that A cup B | vedclass

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Question

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 \c

Vedclass · Accepted Answer

Vedclass · Answer

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, $x \in B$

$	herefore x \in A \cap B$

$\Rightarrow x \in A \cap C \quad[\because A \cap B=A \cap C]$

$	herefore x \in A$ and $x \in C$

$	herefore x \in C$

$	herefore B \subset C$

Similarly, we can show that $C \subset B$

$	herefore B=C$

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 $B = C$

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