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How can I prove the following?

If $A$ is a tautology and $A\implies B$ is a tautology, then $B$ is a tautology.

Nitin Uniyal
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jjAbrahams1
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2 Answers2

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Suppose $A$ is a tautology and $A \Rightarrow B$ is a tautology. Is there a truth-value assignment $v$ that would make $B$ false ? If so, then, since $A \Rightarrow B$ is a tautology, for this $v$ we would have that $A$ should be false. But then we have found a truth-value assignment for which $A$ is false. This contradicts the assumption that $A$ is a tautology.

Hans Hüttel
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  • Ok but how can I prove it. The sentence that I wrote is a true sentence I have to prove that – jjAbrahams1 Oct 28 '16 at 10:40
  • My answer is a proof by contradiction of a statement "if $P$ then $Q$". I assume $P$ holds but also that $Q$ does not hold -- and use this to reach a contradiction. The contradiction tells us that the assumption, namely that $Q$ does not hold, cannot be true. – Hans Hüttel Oct 28 '16 at 10:43
  • oh yes, that is true – jjAbrahams1 Oct 28 '16 at 10:49
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$A$ is a tautology means that $A$ is always true.

So if $A$ is true and $A\implies B$ is true, then $B$ is always true.

So $B$ is a tautology.

E. Joseph
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