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1
GATE CSE 2018
MCQ (Single Correct Answer)
+2
-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?

A
There exists at least one model of $$\varphi $$ with universe of size less than or equal to $$3.$$
B
There exists no model of $$\varphi $$ with universe of size less than or equal to $$3.$$
C
There exists no model of $$\varphi $$ with universe of size greater than $$7.$$
D
Every model of $$\varphi $$ has a universe of size equal to $$7.$$
2
GATE CSE 2016 Set 2
MCQ (Single Correct Answer)
+2
-0.6
Which one of the following well-formed formulae in predicate calculus is NOT valid?
A
$$\left( {\forall xp\left( x \right) \vee \forall xq\left( x \right)} \right) \Rightarrow \left( {\exists x\neg p\left( x \right) \vee \forall xq\left( x \right)} \right)$$
B
$$\left( {\exists xp\left( x \right) \vee \exists xq\left( x \right)} \right) \Rightarrow \exists x\left( {p\left( x \right) \vee q\left( x \right)} \right)$$
C
$$\exists x\left( {p\left( x \right) \wedge q\left( x \right)} \right) \Rightarrow \left( {\exists xp\left( x \right) \wedge \exists xq\left( x \right)} \right)$$
D
$$\forall x\left( {p\left( x \right) \vee q\left( x \right)} \right) \Rightarrow \left( {\forall xp\left( x \right) \vee \forall xq\left( x \right)} \right)$$
3
GATE CSE 2015 Set 2
MCQ (Single Correct Answer)
+2
-0.6
Which one of the following well formed formulae is a tautology?
A
$$\forall x\,\exists y\,R\left( {x,y} \right) \leftrightarrow \exists y\forall x\,R\left( {x,y} \right)$$
B
$$\left( {\forall x\left[ {\exists y\,R\left( {x,y} \right) \to S\left( {x,y} \right)} \right]} \right) \to \forall x\exists y\,S\left( {x,y} \right)$$
C
$$\left[ {\forall x\,\exists y\,\left( {P\left( {x,y} \right)} \right. \to R\left( {x,y} \right)} \right] \leftrightarrow \left[ {\forall x\,\exists y\,\left( {\neg P\left( {x,y} \right)V\,R\left( {x,y} \right)} \right.} \right]$$
D
$$\forall x\,\forall y\,P\left( {x,y} \right) \to \forall x\forall y\,P\left( {y,x} \right)$$
4
GATE CSE 2014 Set 3
MCQ (Single Correct Answer)
+2
-0.6
The CORRECT formula for the sentence, "not all rainy days are cold" is
A
$$\forall d\left( {Rainy\left( d \right) \wedge \sim Cold\left( d \right)} \right)$$
B
$$\forall d\left( { \sim Rainy\left( d \right) \to Cold\left( d \right)} \right)$$
C
$$\exists d\left( { \sim Rainy\left( d \right) \to Cold\left( d \right)} \right)$$
D
$$\exists d\left( {Rainy\left( d \right) \wedge \sim Cold\left( d \right)} \right)$$
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Questions Asked from Marks 2
GATE CSE 2025 Set 1 (1) GATE CSE 2020 (1) GATE CSE 2019 (1) GATE CSE 2018 (1) GATE CSE 2016 Set 2 (1) GATE CSE 2015 Set 2 (1) GATE CSE 2014 Set 3 (1) GATE CSE 2014 Set 2 (1) GATE CSE 2014 Set 1 (1) GATE CSE 2013 (2) GATE CSE 2011 (1) GATE CSE 2010 (1) GATE CSE 2009 (3) GATE CSE 2008 (5) GATE CSE 2007 (1) GATE CSE 2006 (5) GATE CSE 2005 (3) GATE CSE 2004 (2) GATE CSE 2003 (1) GATE CSE 2000 (1) GATE CSE 1996 (1) GATE CSE 1995 (1) GATE CSE 1994 (1) GATE CSE 1990 (1)
GATE CSE Subjects
Theory of Computation
Operating Systems
Algorithms
Digital Logic
Database Management System
Data Structures
Computer Networks
Software Engineering
Compiler Design
Web Technologies
General Aptitude
Discrete Mathematics
Programming Languages
Computer Organization