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Thread: Proof of the subset equality (using Boolean algebra logic)

  1. #1
    Member Pranas's Avatar
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    Proof of the subset equality (using Boolean algebra logic)

    Using Boolean algebra logic I am asked to prove

    $\displaystyle \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.
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    Quote Originally Posted by Pranas View Post
    Using Boolean algebra logic I am asked to prove $\displaystyle \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 $\displaystyle x\in A$ then $\displaystyle x\in B$ is equivalent to If $\displaystyle x\notin B$ then $\displaystyle x\notin A$.
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