Conditional iff Biconditional of Consequent with Disjunction

Theorem

$p \implies q \dashv \vdash q \iff \paren {p \lor q}$


Proof

We apply the Method of Truth Tables.

As can be seen by inspection, the appropriate truth values match for all boolean interpretations.

$\begin{array}{|ccc||ccccc|} \hline p & \implies & q & q & \iff & (p & \lor & q) \\ \hline \F & \T & \F & \F & \T & \F & \F & \F \\ \F & \T & \T & \T & \T & \F & \T & \T \\ \T & \F & \F & \F & \F & \T & \T & \F \\ \T & \T & \T & \T & \T & \T & \T & \T \\ \hline \end{array}$

$\blacksquare$


Sources

  • 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.6$
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