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

| p | q | $q\rightarrow p$ | LHS: $p\rightarrow(q\rightarrow p)$ | $\sim p$ | $p\rightarrow q$ | RHS: $\sim p\rightarrow(p\rightarrow q)$ | |---|---|---|---|---|---|---| | T | T | T | T | F | T | T | | T | F | T | T | F | F | T | | F | T | F | T | T | T | T | | F | F | T | T | T | T | T | …

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

Exercise 1.2 · Q56

p \rightarrow (q \rightarrow p) \equiv \sim p \rightarrow (p \rightarrow q)

Maharashtra MsbshseTextbookSubjectiveImportance★★★★★

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

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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 truth tables, prove the following logical equivalence: $p \rightarrow (q \rightarrow p) \equiv \sim p \rightarrow (p \rightarrow q)$

Concept understanding — Logical Equivalence

Q.Using truth tables, prove the following logical equivalence: p→(q→p)≡∼p→(p→q)p \rightarrow (q \rightarrow p) \equiv \sim p \rightarrow (p \rightarrow q)p→(q→p)≡∼p→(p→q)

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

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