Using truth tables, prove the following logical equivalence: sim p land q ≡ (p lor q) land sim p

Exercise 1.2 · Q53

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

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p	q	$\sim p$	LHS: $\sim p\land q$	$p\lor q$	RHS: $(p\lor q)\land\sim p$
T	T	F	F	T	F
T	F	F	F	T	F
F	T	T	T	T	T
F	F	T	F	F	F

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

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p\leftrightarrow q

›Reveal solution

Build the truth table for both statement patterns for every combination of $p,q$ and compare the final columns.

$p$	$q$	$\sim p$	$\sim q$	$p\leftrightarrow q$	$p\wedge q$	$\sim p\wedge\sim q$	$(p\wedge q)\vee(\sim p\wedge\sim q)$
T	T	F	F	T	T	F	T
T	F	F	T	F	F	F	F
F	T	T	F	F	F	F	F
F	F	T	T	T	F	T	T

The column for $p\leftrightarrow q$ is $[T, F, F, T]$ and the column for $(p\wedge q)\vee(\sim p\wedge\sim q)$ is also $[T, F, F, T]$ .

…

$p$	$q$	$p\wedge q$	$\sim q$	$(p\wedge q)\vee\sim q$	$p\vee\sim q$
T	T	T	F	T	T
T	F	F	T	T	T
F	T	F	F	F	F
F	F	F	T	T	T
…

Q.Using truth tables, prove the following logical equivalence: $\sim p \land q \equiv (p \lor q) \land \sim p$