Using the truth table, prove the following logical equivalence: $p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$.

> > [!TLDR] > Build the truth table for all four combinations of $p,q$ and compare the final columns. > > | $p$ | $q$ | $p\leftrightarrow q$ | $p\wedge q$ | $\sim p$ | $\sim q$ | $\sim p\wedge\sim q$ | $(p\wedge q)\vee(\sim p\wedge\sim q)$ | > |---|---|---|---|---|---|---|---| > | T | T | T | T | F | F | F | T | > | T | F | F | F | F | T | F | F | > | F | T | F | F | T | F | F | F | > | F | F | T | F | T | T | T | T | > > The column for $p\leftrightarrow q$ reads (T, F, F, T) and the column for $(p\wedge q)\vee(\sim p\wedge\sim q)$ also reads (T, F, F, T) — identical in every row. > …

Using the truth table, prove the following logical equivalence: p rightarrow q ≡ (p wedge q) vee (sim p wedge

Question 202 of 227

p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2018Subjective· 3mImportance★★★★★

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Build the truth table for all four combinations of $p,q$ and compare the final columns.

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

The column for $p\leftrightarrow q$ reads (T, F, F, T) and the column for $(p\wedge q)\vee(\sim p\wedge\sim q)$ also reads (T, F, F, T) — identical in every row.

…

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

CBSE 2025Set ANNUAL2 marks

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 $[T, F, F, T]$ .

…

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CBSE 2022Set ANNUAL2 marks
$(p \wedge q) \vee \sim q \equiv p \vee \sim q$
›Reveal solutionSolution

Construct the truth table for both sides and compare all rows.

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

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

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Q.Using the truth table, prove the following logical equivalence: $p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$ .

Concept understanding — Logical Equivalence

Q.Using the truth table, prove the following logical equivalence: p↔q≡(p∧q)∨(∼p∧∼q)p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)p↔q≡(p∧q)∨(∼p∧∼q).

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Q.Using the truth table, prove the following logical equivalence: $p \leftrightarrow q \equiv (p \wedge q) \vee (\sim p \wedge \sim q)$ .

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