Logic, First-Order Representation and Entailment Previous Year Questions (PYQs) for GATE DA: 4+ Solved Questions with Step-by-Step Solutions

Solve 4+ Logic, First-Order Representation and Entailment previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Logic, First-Order Representation and Entailment

Chapter Roadmap

Knowledge Representation and Reasoning

First-Order Logic Translation & Quantifier Validity

Translating English to FOL, scope of quantifiers, and validity checks.

Logical Entailment, Implication & Predicate Reasoning

When does X imply Y? Inference rules and proof strategies.

Why this matters: Exam questions often test your ability to spot invalid translations or correct entailments. Mastery here prevents errors in later reasoning topics.

Topic Hero: From English to First-Order Logic

The Core Translation Pattern

First-Order Logic extends propositional logic by introducing objects, predicates, and quantifiers.

1. The Universal Quantifier ( $\forall$ )

Used for "All", "Every", "Each".

Standard Form: $\forall x (P (x) ⟹ Q (x))$

Example: "All dogs are mammals."

\forall x (Dog (x) ⟹ Mammal (x))

Note: Using

\land

here is a common trap.

\forall x (Dog (x) \land Mammal (x))

means "Everything in the universe is a dog and a mammal," which is false.

2. The Existential Quantifier ( $\exists$ )

Used for "Some", "There exists", "At least one".

Standard Form: $\exists x (P (x) \land Q (x))$

Example: "Some dogs are brown."

\exists x (Dog (x) \land Brown (x))

Note: Using

⟹

here is a trap.

\exists x (Dog (x) ⟹ Brown (x))

is true if there is even one non-dog in the universe (because False

⟹

Anything is True).

Quantifier	Connective	Why?
$\forall$	$⟹$	We restrict the property to a subset.
$\exists$	$\land$	We assert the existence of an object with both properties.

Logic, First-Order Representation and Entailment: Solved Questions with Step-by-Step Explanations (4 Problems)

Question 1 · Artificial Intelligence MCQ

Which of the following statements is NOT true?
(The names of the predicates are intuitive.)

A.
$\forall x \forall y C l a ss ma t e (x, y) \Rightarrow C l a ss ma t e (y, x)$
B.
$\forall x L ik es (x, I cecr e am) \Rightarrow \neg\exists x \neg L ik es (x, I cecr e am)$
C.
“Each king is a person” is equivalent to $\forall x I sK in g (x) \land I s P er so n (x)$
D.
“All humans are mortal” is equivalent to $\forall x I sH u man (x) \Rightarrow I s M or t a l (x)$

Question 2 · Artificial Intelligence MSQ

Sentence

X

is said to entail Sentence

Y

if whenever

X

T R U E

Y

also must hold

T R U E

.
Which of the following statements is/are correct if

X

entails

Y

A.
$X \Rightarrow Y$
B.
$X \land \neg Y$ is $F A L S E$
C.
if $X$ then $Y$
D.
if $Y$ then $X$

Question 3 · Artificial Intelligence MSQ

Let

P (x)

be a predicate.

Which of the following statements is/are NOT valid in first-order logic?

A.
$\forall x P (x) \Rightarrow \exists x P (x)$
B.
$\exists x P (x) \Rightarrow \forall x P (x)$
C.
$\exists x P (x) \Leftrightarrow \forall x P (x)$
D.
$\forall x P (x) \Rightarrow \exists x \neg P (x)$

Question 4 · Artificial Intelligence MSQ

Let game(ball, rugby) be true if the ball is used in rugby and false otherwise.
Let shape(ball, round) be true if the ball is round and false otherwise.
Consider the following logical sentences:
s1:

\forall

ball

\neg

game(ball, rugby)

⟹

shape(ball, round)
s2:

\forall

ball

\neg

shape(ball, round)

⟹

game(ball, rugby)
s3:

\forall

ball game(ball, rugby)

⟹ \neg

shape(ball, round)
s4:

\forall

ball shape(ball, round)

⟹ \neg

game(ball, rugby)
Which of the following choices is/are logical representations of the assertion,
“All balls are round except balls used in rugby”?

A.
$s 1 \land s 3$
B.
$s 1 \land s 2$
C.
$s 2 \land s 3$
D.
$s 3 \land s 4$

A question from this chapter

Question 1

Which of the following statements is NOT true? (The names of the predicates are intuitive.)

Question 2

Sentence X is said to entail Sentence Y if whenever X is T R U E, Y also must hold T R U E . Which of the following statements is/are correct if X entails Y ?

Question 3

Let P (x) be a predicate. Which of the following statements is/are NOT valid in first-order logic?

Question 4

Let game(ball, rugby) be true if the ball is used in rugby and false otherwise. Let shape(ball, round) be true if the ball is round and false otherwise. Consider the following logical sentences: s1: \forall ball \neg game(ball, rugby) ⟹ shape(ball, round) s2: \forall ball \neg shape(ball, round) ⟹ game(ball, rugby) s3: \forall ball game(ball, rugby) ⟹ \neg shape(ball, round) s4: \forall ball shape(ball, round) ⟹ \neg game(ball, rugby) Which of the following choices is/are logical representations of the assertion, “All balls are round except balls used in rugby”?

Free preview ends here

Login to view the complete previous-year questions and solutions

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free