First slide

Operations on Sets

Question

If $X, Y$ and $A$ are three sets such that $A \cap X = A \cap Y$ and $A \cup X = A \cup Y$ then

Moderate

A

$X \subset Y$
B

$Y \subset X$
C

$X = Y$
D

none of these

Solution

If $X \subset Y,$ let $y \in Y$ and $y \notin X$ then either $y \notin A .$ or $y \in A$
So if $y \in A$ , then $y \in A$ $\cap Y$ but $y \notin A \cap X$ and thus $A \cap X \neq A \cap Y .$
If $y \notin A,$ then $y \in A$ $\cap Y$ but $y \notin A$ $\cup X$
and thus $A \cup X \neq A \cup Y$
So $X \subseteq Y,$ similarly $Y \subseteq X .$
$\Rightarrow X = Y .$

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