First slide

Introduction to mathematical reasoning

Question

Let $p$ and $q$ be two statements, then $q \leftrightarrow (~ p \lor ~ q)$ is logically equivalent to

Easy

A

$p$
B

$q$
C

$p \to q$
D

$~ p \land q$

Solution

$q \leftrightarrow (~ p \lor ~ q) \begin{array}{l} \equiv q \leftrightarrow\sim (p \land q) \\ \equiv [q \to\sim (p \land q)] \land [~ (p \land q) \to q] \\ \equiv [~ q \lor ~ (p \land q)] \land [(p \land q) \lor q] \\ \equiv\sim [q \land (p \land q)] \land q \\ \equiv\sim (p \land q) \land q \\ \equiv (~ p \lor ~ q) \land q \\ \equiv (~ p \land q) \lor (~ q \land q) \\ \equiv (~ p \land q) \end{array}$

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