If A and B are two sets,then A cup B = A cap | vedclass

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Question

(C) Given $A \cup B = A \cap B$.Let $x \in A$. Since $A \subseteq A \cup B$,it follows that $x \in A \cup B$.Because $A \cup B = A \cap B$,we have $x

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$A = B$

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(C) Given $A \cup B = A \cap B$.Let $x \in A$. Since $A \subseteq A \cup B$,it follows that $x \in A \cup B$.Because $A \cup B = A \cap B$,we have $x \in A \cap B$.By the definition of intersection,$x \in A \cap B \implies x \in A$ and $x \in B$. Thus,$x \in B$.Since $x \in A \implies x \in B$,we conclude $A \subseteq B$.Similarly,let $y \in B$. Since $B \subseteq A \cup B$,it follows that $y \in A \cup B$.Because $A \cup B = A \cap B$,we have $y \in A \cap B$.By the definition of intersection,$y \in A \cap B \implies y \in A$ and $y \in B$. Thus,$y \in A$.Since $y \in B \implies y \in A$,we conclude $B \subseteq A$.Since $A \subseteq B$ and $B \subseteq A$,it must be that $A = B$.

If $A$ and $B$ are two sets,then $A \cup B = A \cap B$ if and only if:

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