First slide

Introduction to mathematical reasoning

Question

The only statement among the following that is a tautology is:

Easy

A

$A \land (A \lor B)$
B

$A \lor (A \land B)$
C

$[A \land (A \to B)] \to B$
D

$B \to [A \land (A \to B)]$

Solution

Note that
$A \land (A \lor B)$ is $F$ when $A = F,$
$A \lor (A \land B)$ is $F$ when $A = F, B = F,$
and $B \to [A \land (A \to B)]$ is $F$ when $A = F, B = T$
$∴$ We check only (c)
$\begin{array}{l} [A \land (A \to B)] \to B \\ \equiv [A \land (~ A \lor B)] \to B \\ \equiv [(A \land (~ A))] \lor (A \land B)] \to B \\ \equiv A \land B \to B \\ \equiv ~ (A \land B) \lor B \equiv\sim [(A \land B) \land (~ B)] \\ \equiv ~ [A \land (B \land ~ B)] \equiv\sim [A \land F] \equiv\sim F \equiv T \end{array}$
Thus,
$[A \land (A \to B)] \to B$ is a tautology.

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