Question states:
Prove the law of double complements for sets:
If $A$ is a set and $A^\complement$ is its complement than prove that:
$$ (A^\complement)^\complement = A$$
I started with:
$$ A^\complement = U – A$$
where U is the universal set.
But I do not understand how to go further.
Best Answer
Let $A$ be a subset of some universe $U$.
We'll prove this by proving that the two sets are subsets of each other (and must therefore be equal). We'll use nothing but the definition that if $x \in X$, then $x \notin X^C$ (and what clearly follows: if $x \notin X$, then $x \in X^C$).
Take $a \in A$. Then $a \notin A^C$. Then $a \in (A^C)^C$, by definition of the complement. So $A \subseteq (A^C)^C$.
Now take $a \in (A^C)^C$. That means $a \notin A^C$. Therefore $a \in A$. So $(A^C)^C \subseteq A$.
As $A \subseteq (A^C)^C$ and $(A^C)^C \subseteq A$, we conclude that $A = (A^C)^C$.
For other techniques, see here. This is, in some sense, a duplicate question.