GATE CSE 2025 Set 1 | Mathematical Logic Question 2 | Discrete Mathematics

GATE CSE 2025 Set 1

MCQ (More than One Correct Answer)

-0

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:

$\bullet$ mother $(y, x): y$ is the mother of $x$

$\bullet$ noteq $(x, y): x$ and $y$ are not equal

$\forall x \exists y \exists z$ (mother $(y, x) \wedge \neg \operatorname{mother}(z, x))$

$\forall x \exists y[\operatorname{mother}(y, x) \wedge \forall z(\operatorname{noteq}(z, y) \rightarrow \neg \operatorname{mother}(z, x))]$

$\forall x \forall y[\operatorname{mother}(y, x) \rightarrow \exists z(\operatorname{mother}(z, x) \wedge \neg \operatorname{noteq}(z, y))]$

$\forall x \exists y[\operatorname{mother}(y, x) \wedge \neg \exists z(\operatorname{noteq}(z, y) \wedge \operatorname{mother}(z, x))]$

GATE CSE 2020

MCQ (Single Correct Answer)

-0.67

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

Note that W is a predicate formula without any free occurrence of x.

$$\forall x$$(p(x) $$ \vee $$ W) $$ \equiv $$ $$\forall x$$ p(x) $$ \vee $$ W

$$\exists x$$(p(x) $$ \wedge $$ W) $$ \equiv $$ $$\exists x$$ p(x) $$ \wedge $$ W

$$\forall x$$(p(x) $$ \to $$ W) $$ \equiv $$ $$\forall x$$ p(x) $$ \to $$ W

$$\exists x$$(p(x) $$ \to $$ W) $$ \equiv $$ $$\exists x$$ p(x) $$ \to $$ W

GATE CSE 2019

MCQ (Single Correct Answer)

-0.67

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?

S1 and S3

S1, S2 and S3

S2 and S3

S1 and S2

GATE CSE 2018

MCQ (Single Correct Answer)

-0.6

Consider the first-order logic sentence
$$\varphi \equiv \,\,\,\,\,\,\,\exists s\exists t\exists u\forall v\forall w$$ $$\forall x\forall y\psi \left( {s,t,u,v,w,x,y} \right)$$
where $$\psi $$ $$(𝑠,𝑡, 𝑢, 𝑣, 𝑤, 𝑥, 𝑦)$$ is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose $$\varphi $$ has a model with a universe containing $$7$$ elements.

Which one of the following statements is necessarily true?

There exists at least one model of $$\varphi $$ with universe of size less than or equal to $$3.$$

There exists no model of $$\varphi $$ with universe of size less than or equal to $$3.$$

There exists no model of $$\varphi $$ with universe of size greater than $$7.$$

Every model of $$\varphi $$ has a universe of size equal to $$7.$$