If A and B have n elements in common,how many | vedclass

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Question

(C) Let $A \cap B$ be the set of common elements between $A$ and $B$. Given that $A$ and $B$ have $n$ elements in common,the number of elements in $A

Vedclass · Accepted Answer

$n^{2}$

Vedclass · Answer

(C) Let $A \cap B$ be the set of common elements between $A$ and $B$. Given that $A$ and $B$ have $n$ elements in common,the number of elements in $A \cap B$ is $n$,i.e.,$|A \cap B| = n$.An ordered pair $(x, y)$ belongs to $(A 	imes B) \cap (B 	imes A)$ if and only if $(x, y) \in A 	imes B$ and $(x, y) \in B 	imes A$.This implies $x \in A$ and $y \in B$,and also $x \in B$ and $y \in A$.Therefore,$x \in A \cap B$ and $y \in A \cap B$.Since there are $n$ choices for $x$ from $A \cap B$ and $n$ choices for $y$ from $A \cap B$,the total number of such ordered pairs $(x, y)$ is $n 	imes n = n^{2}$.Thus,$A 	imes B$ and $B 	imes A$ have $n^{2}$ elements in common.

If $A$ and $B$ have $n$ elements in common,how many elements do $A \times B$ and $B \times A$ have in common?

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