First slide

Introduction to mathematical reasoning

Question

Which of the following is a contradiction?

Moderate

A

$p \to (q \to p)$
B

$p \to (p v q)$
C

$(p v q) \to (~ p \land ~ q)$
D

$(p v ~ p) \to (q \land ~ q)$

Solution

$a) p \to (q \to p) \equiv p \to (~ q \lor p) \equiv ~ p \lor (~ q \lor p) \equiv (~ p \lor p) \lor ~ q \equiv t \lor ~ q \equiv t$
$b) p \to (p \lor q) \equiv ~ p \lor (p \lor q) \equiv t \lor q \equiv t$
$c) p \lor q \to (~ p \land ~ q) \equiv ~ (p \lor q) \lor (~ (p \lor q)) \equiv ~ (p \lor q)$
$d) (p \lor ~ p) \to (q \land ~ q) \equiv t \to c \equiv (~ t \lor c) \equiv c \lor c \equiv c$
$L H S \geq 1, R H S \leq 1$

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