Propositional Logic and Logical Equivalence PYQs for GATE DA

Solve 3+ Propositional Logic and Logical Equivalence previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Propositional Logic

Chapter Journey: Propositional Logic

Step 1: The Language of Logic

Translating English to Propositional Logic
Conditional and Biconditional Statements
Understanding Necessary vs. Sufficient Conditions

Step 2: The Mechanics of Truth

Truth Tables and Tautologies
Logical Equivalence and Algebraic Simplification
Fundamental Laws of Logic (De Morgan's, Contrapositive)

Step 3: Exam Mastery

Decoding Complex Word Problems
Identifying Equivalent Propositions Quickly
Avoiding Common Logical Fallacies

The Core of Logical Reasoning

What is Propositional Logic?
Propositional logic is the study of propositions (declarative statements that are either True or False) and how they combine using logical connectives.

Symbol	Name	Meaning
$\land$	Conjunction	AND (Both must be true)
$\lor$	Disjunction	OR (At least one must be true)
$\neg$	Negation	NOT (Flips the truth value)
$\to$	Implication	IMPLIES (If... then...)
$⟺$	Biconditional	IF AND ONLY IF (Same truth value)

The Goal:
To take messy, ambiguous English sentences, convert them into precise symbolic logic, and use mathematical rules to determine their truth values or simplify them.

Propositional Logic and Logical Equivalence: Solved Questions with Step-by-Step Explanations (3 Problems)

Question 1 · Analytical Aptitude · 2026 MCQ

Assume that a Creative

(C)

person will Succeed

(S)

if the person is also Disciplined

(D)

, but will not succeed otherwise. Now, consider the following statements:

(i)

C \land S \Leftrightarrow D

(ii)

C \Rightarrow (S ⟺ D)

(iii)

C \Leftrightarrow ((D \Rightarrow S) \lor \neg S)

Which of the following options is correct?

A.
Both (i) and (ii) are TRUE
B.
Only (ii) is TRUE
C.
Both (ii) and (iii) are TRUE
D.
Only (iii) is TRUE

Correct Answer:

Step-by-Step Solution

Insight: "A creative person will succeed if disciplined, but not otherwise" means: if creative, then (succeed iff disciplined).

Exam route: Translate the English sentence step by step. "Creative person will succeed if disciplined" gives $C \to (D \to S)$ . "But not otherwise" adds $C \to (\neg D \to \neg S)$ . Together: $C \to (S ⟺ D)$ . Check which option matches.

Learning route:

Step 1: Parse the sentence structure.

"A Creative (C) person will Succeed (S) if the person is also Disciplined (D), but will not succeed otherwise."

The phrase "but not otherwise" is the key. It means the condition is both necessary and sufficient.

Step 2: Translate the "if" part.

"will succeed if disciplined" $⟹ D \to S$ (within the context of being creative).

Step 3: Translate the "but not otherwise" part.

"will not succeed otherwise" $⟹ \neg D \to \neg S$ (within the context of being creative).

Step 4: Combine 2 and 3.

$(D \to S) \land (\neg D \to \neg S) \equiv S ⟺ D$ .

Step 5: Apply the context.

The whole rule applies to creative people, so: $C \to (S ⟺ D)$ .

Step 6: Evaluate the given statements.

(i) $C \land S ⟺ D$ : This means $(C \land S) ⟺ D$ , which is not equivalent. For example, if $C = F, D = F$ , the original gives $T$ (vacuously), but (i) gives $F ⟺ F = T$ . Wait, let me check $C = F, S = T, D = F$ : original gives $T$ , (i) gives $(F \land T) ⟺ F = F ⟺ F = T$ . Let me check $C = F, S = F, D = T$ : original gives $T$ , (i) gives $(F \land F) ⟺ T = F ⟺ T = F$ . Not equivalent. So (i) is FALSE.

(ii) $C \to (S ⟺ D)$ : This is exactly what we derived. TRUE.

(iii) $C ⟺ ((D \to S) \lor \neg S)$ : Simplify the right side. $(D \to S) \lor \neg S = (\neg D \lor S) \lor \neg S = \neg D \lor (S \lor \neg S) = \neg D \lor T = T$ . So (iii) becomes $C ⟺ T$ , which means $C$ must be true. This is not equivalent to $C \to (S ⟺ D)$ . FALSE.

Answer: Only (ii) is TRUE, which is option B.

Question 2 · Analytical Aptitude · 2025 MCQ

Let

p

and

q

be any two propositions. Consider the following propositional statements.

S_{1} : p \to q, S_{2} : \neg p \land q, S_{3} : \neg p \lor q, S_{4} : \neg p \lor \neg q,

where

\land

denotes conjunction (AND operation),

\lor

denotes disjunction (OR operation), and

\neg

denotes negation (NOT operation). Which one of the following options is correct?
(Note:

\equiv

denotes logical equivalence)

A.
$S_{1} \equiv S_{3}$
B.
$S_{2} \equiv S_{3}$
C.
$S_{2} \equiv S_{4}$
D.
$S_{1} \equiv S_{4}$

Correct Answer:

Step-by-Step Solution

Insight: $p \to q$ is logically equivalent to $\neg p \lor q$ by material implication.

Exam route: Recall the material implication law $p \to q \equiv \neg p \lor q$ . Compare $S_{1} = p \to q$ with $S_{3} = \neg p \lor q$ . They are identical in meaning.

Learning route:

Step 1: Write out each statement clearly.

$S_{1} : p \to q$

$S_{2} : \neg p \land q$

$S_{3} : \neg p \lor q$

$S_{4} : \neg p \lor \neg q$

Step 2: Apply the material implication equivalence to $S_{1}$ .

$p \to q \equiv \neg p \lor q$

Step 3: Compare the result with $S_{3}$ .

$S_{1} \equiv \neg p \lor q \equiv S_{3}$

Step 4: Verify with a truth table to be absolutely sure.

$p$ | $q$ | $S_{1}$ | $S_{3}$

T | T | T | T

T | F | F | F

F | T | T | T

F | F | T | T

The columns for $S_{1}$ and $S_{3}$ match exactly.

Answer: $S_{1} \equiv S_{3}$ , which is option A.

Common trap: Students sometimes confuse $\neg p \lor q$ with $\neg p \land q$ or $\neg p \lor \neg q$ . Remember that implication becomes a disjunction ( $\lor$ ), not a conjunction ( $\land$ ).

Question 3 · Analytical Aptitude · 2024 MSQ

Let

x

and

y

be two propositions. Which of the following statements is a tautology
/are tautologies?

A.
$(\neg x \land y) ⟹ (y ⟹ x)$
B.
$(x \land \neg y) ⟹ (\neg x ⟹ y)$
C.
$(\neg x \land y) ⟹ (\neg x ⟹ y)$
D.
$(x \land \neg y) ⟹ (y ⟹ x)$

Correct Answer:

["B","C","D"]

Step-by-Step Solution

Insight: Convert each implication to disjunction form and simplify. A tautology simplifies to $T$ .

Exam route: For each option, replace $A \to B$ with $\neg A \lor B$ , then simplify using De Morgan's and absorption. If the result is $T$ , it's a tautology.

Learning route:

Step 1: Recall $A \to B \equiv \neg A \lor B$ .

Step 2: Evaluate option A: $(\neg x \land y) \to (y \to x)$ .

$= \neg (\neg x \land y) \lor (\neg y \lor x)$

$= (x \lor \neg y) \lor (\neg y \lor x)$

$= x \lor \neg y$

This is NOT always true (false when $x = F, y = T$ ). So A is not a tautology.

Step 3: Evaluate option B: $(x \land \neg y) \to (\neg x \to y)$ .

$= \neg (x \land \neg y) \lor (x \lor y)$

$= (\neg x \lor y) \lor (x \lor y)$

$= \neg x \lor y \lor x \lor y$

$= (\neg x \lor x) \lor (y \lor y)$

$= T \lor y$

$= T$

This IS a tautology.

Step 4: Evaluate option C: $(\neg x \land y) \to (\neg x \to y)$ .

$= \neg (\neg x \land y) \lor (x \lor y)$

$= (x \lor \neg y) \lor (x \lor y)$

$= x \lor \neg y \lor x \lor y$

$= x \lor (\neg y \lor y)$

$= x \lor T$

$= T$

This IS a tautology.

Step 5: Evaluate option D: $(x \land \neg y) \to (y \to x)$ .

$= \neg (x \land \neg y) \lor (\neg y \lor x)$

$= (\neg x \lor y) \lor (\neg y \lor x)$

$= (\neg x \lor x) \lor (y \lor \neg y)$

$= T \lor T$

$= T$

This IS a tautology.

Answer: B, C, D are tautologies.

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Propositional Logic and Logical Equivalence PYQs for GATE DA

Chapter Roadmap: Propositional Logic

Chapter Journey: Propositional Logic

The Core of Logical Reasoning

Propositional Logic and Logical Equivalence: Solved Questions with Step-by-Step Explanations (3 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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