First slide

Introduction to mathematical reasoning

Question

Which of the following is always true?

Easy

A

$(~ p \lor ~ q) \equiv (p \land q)$
B

$(p \to q) \equiv (~ q \to\sim p)$
C

$~ (p \to\sim q) \equiv (p \land ~ q)$
D

$~ (p \leftrightarrow q) \equiv (p \to q) \to (q \to p)$

Solution

$Since, ~ (p \lor q) \equiv (~ p \land ~ q) and ~ (p \land q) \equiv (~ p \lor q)$
So,. option (b) and (d) are not true.
$(p \to q) \equiv p \land ~ q$ so option (c) is not true.
$\begin{array}{l} p \to q ~ p \lor q \\ ~ q \to\sim p \equiv [~ (~ q) \lor ~ p] \equiv q \lor ~ p \equiv\sim p \lor q \\ p \to q \equiv\sim q \to\sim p \end{array}$

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