First slide

Introduction to mathematical reasoning

Question

$Let p, q, r be three mathematical statements. The statement (~ p \to q) \to r is equivalent to$

Moderate

A

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

$(~ p \lor r) \land (~ q \lor r)$
C

$(~ p \lor r) \lor (q \lor r)$
D

$a tautology$

Solution

$\begin{array}{l} Consider (~ p \to q) \to r [w e k n o w t h a t (p \to q) = ~ p \lor q] \\ = (~ (~ p) \lor q) \to r = (p \lor q) \to r =\sim (p \lor q) \lor r \\ = (~ p \land ~ q) \lor r \\ = (~ p \lor r) \land (~ q \lor r) \end{array}$
Therefore, the correct answer is (2).

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