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Let A and B be two subsets of some universal set. Prove that...

KOO

New member
Oct 19, 2013
19
**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 $B\subseteq A$?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,904
**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 $B\subseteq A$?
Start with "Let $x\in B$."
Then continue with the same argument you have - just with $A$ and $B$ swapped around...
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
Or...just note that both expressions are symmetric in A and B, and union is commutative...