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

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

Miscellaneous Exercise 1 · Q175

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

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Construct a joint truth table over p, q for both sides.

p=T,q=T: LHS p↔q=T. RHS: p∧q=T; ~p∧~q=F∧F=F; (p∧q)∨(~p∧~q)=T∨F=T. LHS=RHS=T.

p=T,q=F: LHS p↔q=F. RHS: p∧q=F; ~p∧~q=F∧T=F; disjunction=F. LHS=RHS=F.

p=F,q=T: LHS p↔q=F. RHS: p∧q=F; ~p∧~q=T∧F=F; disjunction=F. LHS=RHS=F.

p=F,q=F: LHS p↔q=T. RHS: p∧q=F; ~p∧~q=T∧T=T; disjunction=T. LHS=RHS=T. …

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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
…