Proof of the subset equality (using Boolean algebra logic)

**Pranas** · April 4th 2011, 07:34 AM

Using Boolean algebra logic I am asked to prove

$\displaystyle \[A \subseteq B \Rightarrow \overline B \subseteq \overline A \]$

In mathematical analysis course that looks like plain and simple modus tollens principle, but using Boolean algebra logic I just don't know...

Maybe you can help me?

Thanks in advance.

**Plato** · April 4th 2011, 08:02 AM

Originally Posted by Pranas

Using Boolean algebra logic I am asked to prove $\displaystyle \[A \subseteq B \Rightarrow \overline B \subseteq \overline A \]$
In mathematical analysis course that looks like plain and simple modus tollens principle, but using Boolean algebra logic I just don't know...

That is just the contra-positive of a statement.
"If P then Q" is true then "If not Q then not P" is also true.
If $x\in A$ then $x\in B$ is equivalent to If $x\notin B$ then $x\notin A$ .

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