Stuck with proof techniques I have to prove the following question,

Let A and B are subsets of a universal set U. Prove that A is a subset of
  B iff B' is a subset of  A'

Now I don't understand how do I prove this using proof techniques. Please guide me.
Thanks.
 A: We want to prove the following:
$$A\subseteq B\iff B^c\subseteq A^c$$
We need, therefore, to prove two implications.


*

*Assume $A\subseteq B$. We want to show that $B^c\subseteq A^c$, that is $x\in B^c$ implies $x\in A^c$. Suppose $x\in B^c$, then $x\notin B$. Since $A\subseteq B$ is to say $x\in A\rightarrow x\in B$, we have that if $x\notin B$ then $x\notin A$. In particular, $x\in A^c$. We therefore proved $\Rightarrow$.

*Suppose now that $B^c\subseteq A^c$. Note that $(B^c)^c = B$. By the previous proof we have that $B^c\subseteq A^c\Rightarrow (A^c)^c\subseteq (B^c)^c$. However that means exactly that $A\subseteq B$.

Another approach is to use logical equivalence of $\alpha\rightarrow\beta\iff\lnot\beta\rightarrow\lnot\alpha$.
Since $A\subseteq B$ is $x\in A\rightarrow x\in B$, this as the above equivalence tells us, is the same as $x\notin B\rightarrow x\notin A$, which is precisely to say $B^c\subseteq A^c$.
A: See my formal proof at http://www.dcproof.com/MathSE-2011-09-30.htm
This proof was generated with the aid of my DC Proof software available at
http://www.dcproof.com
Dan
