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Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**

Attempt:

Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore $x\in B$. Since $x$ was an arbitrary element of $A$, this shows that $A\subseteq B$.

How do we show $B\subseteq A$?