# Explain why the following are true for arbitrary statements P and Q:

Explain why the following are true for arbitrary statements P and Q: If P is a tautology then (P or Q) ≡ P

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$P\rightarrow (P\lor Q)$ is a tautology. Truth tables show it immediately; thinking about it shows it in about two and a half instants. And equivalences give it more slowly: $$P\Rightarrow (P\lor Q) \equiv \neg P\lor (P\lor Q) \equiv (\neg P\lor P)\lor Q \equiv \neg P\lor P.$$
So whenver $P$ is true, $P\lor Q$ is also true.
So if $P$ is always true, then $P\lor Q$ is always true.
So if $P$ is a tautology, then $P\lor Q$ is a tautology; and two tautologies are always equivalent.