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Definition (Implication)
If $P$ and $Q$ are statements or open sentences, then an implication, written symbolically as $P\implies Q$, is the sentence "If $P$, then $Q$" or equivalently "$P$ implies $Q$". There are several equivalent ways to express this sentence such as "$Q$ if $P$" or "$P$ only if $Q$." The implication $P\implies Q$ is true unless $P$ is true and $Q$ is false.