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Equivalence

Definition 2.3.17   Let $ p$ and $ q$ be two propositions. The equivalence $ p
\Longleftrightarrow q$ is true whenever $ p$ and $ q$ have the same logical value.
$ p$ $ q$ $ p
\Longleftrightarrow q$
T T T
T F F
F T F
F F T

Proposition 2.3.18   For any two propositions $ p$ and $ q$ , $ p
\Longleftrightarrow q$ is logically equivalent to $ (p \Longrightarrow q) \wedge (q
\Longrightarrow p)$.

Proof. We build a truth table:
$ p$ $ q$ $ p
\Longleftrightarrow q$ $ p
\Longrightarrow q$ $ q \Longrightarrow p$ $ (p \Longrightarrow q) \wedge (q
\Longrightarrow p)$
T T T T T T
T F F F T F
F T F T F F
F F T T T T
As the third column and the last column are identical, the claim is true. $ \qedsymbol$



root 2002-06-10