Which of the following is a fallacy?

$(p \land q) \land \sim (p \lor q)$

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$p \lor (\sim p \land q)$

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$(p \Rightarrow q) \Rightarrow p$

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None of these

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Solution

The correct option is A
$(p \land q) \land \sim (p \lor q)$

$(p \land q) \land \sim (p \lor q) = (p \land q) \land (\sim p \land \sim q) [∵ \sim (p \lor q) = \sim p \land \sim q] = (p \land \sim p) \land (q \land \sim q)$
$(p \land \sim p)$ and $(q \land \sim q)$ are always false.
$∴ (p \land \sim p) \land \sim (p \lor q)$ is false for all values of p and q.
Hence, it is a fallacy.

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Question 7 (b)

Identify like terms in the following:

$10 p q, 7 p, 8 q, - p^{2} q^{2}, - 7 q p, - 100 q, - 23, 12 q^{2} p^{2}, - 5 p^{2}, 41, 2405 p, 78 q p, 13 p^{2} q, q p^{2}, 701 p^{2}$

(S_{1}) : (q \lor p) \to (p \leftrightarrow\sim q)

(S_{2}) :\sim q \land (\sim p \leftrightarrow q)

x^{2} + p x + q = 0

Factorise: p^{2} + q^{2} + 2 (p q + q r + r p) .

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