Double Complement of a set proof
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.
$\endgroup$ 03 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$I know it's old, but here's how I solved it using logical equivalences. For people who coming from the future, I was required to solve it like this.
We know that: $A=\{x|x\in A\}$, which reads: there is an x in the set of A. And we know that the complement of A is $A^c=\{x| x\notin A\}$, which reads: there is an x which is not in A.
We start with $(A^c)^c = \{x|x\notin A^c\}$ is read as: there is an x that is not in the complement set of A. That means that if an x is not in the complement set of A (and the complement set of A is the set that doesn't have A), then it must be in A. All we do is use negation rules.
$(A^c)^c = \{x|x\notin A^c\}$
$(A^c)^c = \{x|\neg (x \in A^c)\}$
$(A^c)^c = \{x|\neg (x \notin A)\}$
$(A^c)^c = \{x|\neg (\neg (x \in A))\}$
$(A^c)^c = \{x| x \in A\}$
Therefore
$(A^c)^c=A$
$\endgroup$ $\begingroup$I am not completely sure of how valid my proof is, but i'd like to share it, if someone would like to tell me what's wrong about the proof.
$A^C = U - A$
$(A^C)^C = U - (A^C)$
And now substituting the first statement.
$(A^C)^C = U - (U - A)$
$(A^C)^C = U - U + A = A$
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