Mathematical Logics | GATE PYQ [2010-2025]

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Question 1

Which one of the following well-formed formulae in predicate calculus is NOT valid? [ MCQ || GATE 2016 Set-2 || 2 Marks ]

• (∀x p(x) ⇒∀x q(x)) ⇒ (∃x ¬p(x) ∨ ∀x q(x))

• (∃x p(x) ∨ ∃x q(x)) ⇒∃x (p(x) ∨ q(x))

• ∃x (p(x) ∧ q(x)) ⇒ (∃x p(x) ∧ ∃x q(x))

• ∀x (p(x) ∨ q(x)) ⇒ (∀x p(x) ∨ ∀x q(x))

Question 2

Let p and q be the following propositions:

• p: Fail grade can be given.
• q: Student scores more than 50% marks.

Consider the statement: “Fail grade cannot be given when student scores more than 50% marks.”

Which one of the following is the CORRECT representation of the above statement in propositional logic? [ MCQ || GATE 2024 Set-2 || 1 Marks ]

• q→¬p

• q→p

• p→q

• ¬p→q

Question 3

Let P(x) be an arbitrary predicate over the domain of natural numbers.
Which ONE of the following statements is TRUE? [ MCQ || GATE 2025 Set-2 || 1 Marks ]

• (P(0) ∧ ∀x[P(x)⇒P(x+1)]) ⇒ (∀xP(x))

• (P(0) ∧ ∀x[P(x) ⇒ P(x−1)]) ⇒ (∀xP(x))

• (P(1000) ∧ ∀x[P(x) ⇒ P(x−1)]) ⇒ (∀xP(x))

• (P(1000)∧∀x[P(x) ⇒P(x+1)]) ⇒ (∀xP(x))

Question 4

Consider two relations describing teams and players in a sports league:

• teams(tid, tname): tid and tname are team-id and team-name, respectively.
• players(pid, pname, tid): pid, pname, and tid denote player-id, player-name, and the team-id of the player, respectively.

Which ONE of the following tuple relational calculus queries returns the name of the players who play for the team having tnametnametname as 'MI'? [ MCQ || GATE 2025 Set-1 || 2 Marks ]

• {p.pname ∣ p∈players ∧ ∃t(t∈teams ∧ p.tid=t.tid∧t.tname=′MI′)}

• {p.pname ∣ p∈teams∧∃t(t∈players ∧ p.tid=t.tid ∧ t.tname=′MI′)}

• {p.pname ∣ p∈players ∧ ∃t (t∈teams ∧ t.tname=′MI′)}

• {p.pname ∣ p∈teams ∧ ∃t(t∈players ∧ t.tname=′MI′)}

Question 5

Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement:
“Everyone has exactly one mother”?

The meanings of the predicates used are:

• mother(y,x): y is the mother of x
• noteq(x,y): x and y are not equal [ MSQ || GATE 2025 Set-1 || 2 Marks ]

• ∀x∃y∃z(mother(y,x)∧¬mother(z,x))

• ∀x∃y[mother(y,x)∧∀z(noteq(z,y)→¬mother(z,x))]

• ∀x∀y[mother(y,x)→∃z(mother(z,x)∧¬noteq(z,y))]

• ∀x∃y[mother(y,x)∧¬∃z(noteq(z,y)∧mother(z,x))]

Question 6

Geetha has a conjecture about integers, which is of the form
∀x(P(x) ⇒ ∃yQ(x, y)),
where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha’s conjecture? [ MSQ || GATE 2023 || 1 Marks ]

• ∃x(P(x) ∧ ∀yQ(x, y))

• ∀x∀yQ(x, y)

• ∃y∀x(P(x) ⇒ Q(x, y))

• ∃x(P(x) ∧ ∃yQ(x, y))

Question 7

Choose the correct choice(s) regarding the following propositional logic assertion

S: ((P ∧ Q) → R) → ((P ∧ Q) → (Q → R))

Which of the following is true? [ MSQ || GATE 2021 Set-2 || 1 Marks ]

• S is neither a tautology nor a contradiction.

• S is a tautology.

• S is a contradiction.

• The antecedent of S is logically equivalent to the consequent of S.

Question 8

Let p and q be two propositions. Consider the following two formulae in propositional logic.

S₁:¬p ∧ ( p ∨ q )→q

S₂: q → (∼p ∧ (p ∨ q))

Which one of the following choices is correct? [ MCQ || GATE 2021 Set-1 || 1 Marks ]

• Neither S₁ nor S₂ is a tautology.

• S₁ is not a tautology but S₂ is not a tautology.

• S₁ is a tautology but S₂ is not a tautology.

• Both S₁ and S₂ are tautologies.

Question 9

Which one of the following predicate formulae is NOT logically valid?

Note that W is a predicate formula without any free occurrence of x. [ MCQ || GATE 2020 || 2 Marks ]

• ∃x(p(x) → W) ≡ ∀xp(x) → W

• ∀x(p(x) → W) ≡ ∀x p(x) → W

• ∃x(P(x) ∧ W) ≡ ∃x P(x) ∧ W

• ∀x(p(x) ∨ W) ≡ ∀ xp(x) ∨ W

Question 10

Consider the first order predicate formula φ: ∀x[(∀z z|x⇒ ((z = x) ∨ (z = 1))) ⇒∃w (w >x) ∧ (∀z z|w⇒ ((w = z) ∨ (z = 1)))]

Here 'a|b' denotes that 'a divides b' where a and b are integers. Consider the following sets:

S1. {1, 2, 3, ..., 100}

S2. Set of all positive integers

S3. Set of all integers

Which of the above sets satisfy φ? [ MCQ || GATE 2019 || 2 Marks ]

• S1 and S3

• S1, S2 and S3

• S2 and S3

• S1 and S2

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Discrete Mathematics