- #1
KOO
- 19
- 0
**Let A and B be two subsets of some universal set.
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 $B\subseteq A$?
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 $B\subseteq A$?