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GATE CSE

Mathematical Logic

Discrete Mathematics

Previous Years Questions

Marks 1

More
Consider the following expressions: $$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(i)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ false $$\,...
GATE CSE 2016 Set 2
Let $$p,q,r,s$$ represent the following propositions. $$p:\,\,\,x \in \left\{ {8,9,10,11,12} \right\}$$ $$q:\,\,\,x$$ is...
GATE CSE 2016 Set 1
In a room there are only two types of people, namely Type $$1$$ and Type $$2.$$ Type $$1$$ people always tell the truth ...
GATE CSE 2015 Set 3
Consider the following two statements. $$S1:$$ If a candidate is known to be corrupt, then he will not be elected $$S2:$...
GATE CSE 2015 Set 2
Consider the following statements: P: Good mobile phones are not cheap Q: Cheap mobile phones are not good L: P implies ...
GATE CSE 2014 Set 3
Consider the statement "Not all that glitters is gold" Predicate glitters$$(x)$$ is true if $$x$$ glitters and predic...
GATE CSE 2014 Set 1
The truth table Represents the Boolean function...
GATE CSE 2012
What is the correct translation of the following statement into mathematical logic? "Some real numbers are rational"
GATE CSE 2012
Consider the following logical inferences. $${{\rm I}_1}:$$ If it rains then the cricket match will not be played. The...
GATE CSE 2012
A set of Boolean connectives is functionally complete if all Boolean function can be synthesized using those, Which of t...
GATE CSE 2008
Let $$a(x,y)$$, $$b(x,y)$$ and $$c(x,y)$$ be three statements with variables $$x$$ and $$y$$ chosen from some universe. ...
GATE CSE 2004
Identify the correct translation into logical notation of the following assertion. $$Some\,boys\,in\,the\,class\,are\,t...
GATE CSE 2004
"If X then Y unless Z" is represented by which of the following formulas in propositional logic? (" $$\neg $$ " is negat...
GATE CSE 2002
Consider two well-formed formulas in propositional logic $$F1:P \Rightarrow \neg P$$ $$F2:\left( {P \Rightarrow \neg P}...
GATE CSE 2001
What is the converse of the following assertion? I stay only if you go
GATE CSE 1998
The proposition $$p \wedge \left( { \sim p \vee q} \right)$$ is
GATE CSE 1993
Which of the following predicate calculus statements is/are valid?
GATE CSE 1992

Marks 2

More
Let p and q be two propositions. Consider the following two formulae in propositional logic. S1 : (¬p ∧ (p&nb...
GATE CSE 2021 Set 1
Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any fre...
GATE CSE 2020
Consider the first-order logic sentence $$\varphi \equiv \,\,\,\,\,\,\,\exists s\exists t\exists u\forall v\forall w$$...
GATE CSE 2018
Which one of the following well-formed formulae in predicate calculus is NOT valid?
GATE CSE 2016 Set 2
Which one of the following well formed formulae is a tautology?
GATE CSE 2015 Set 2
Which one of the following Boolean expressions is NOT A tautology?
GATE CSE 2014 Set 2
The CORRECT formula for the sentence, "not all rainy days are cold" is
GATE CSE 2014 Set 3
Which one of the following propositional logic formulas is TRUE when exactly two of $$p, q,$$ and $$r$$ are TRUE?...
GATE CSE 2014 Set 1
Which one of the following is NOT logically equivalent to $$\neg \exists x\left( {\forall y\left( \alpha \right) \wedge...
GATE CSE 2013
What is the logical translation of the following statement? "None of my friends are perfect."
GATE CSE 2013
Which one of the following options is correct given three positive integers $$x, y$$ and $$z$$, and a predicate $$P\lef...
GATE CSE 2011
Suppose the predicate $$F(x,y,t)$$ is used to represent the statements that person $$x$$ can fool person $$y$$ at time ...
GATE CSE 2010
The binary operation ◻ is defined as follows: Which one of the following is equivalence to $$P \vee Q$$? ...
GATE CSE 2009
Consider the following well-formed formulae: $${\rm I}.$$ $$\,\,\neg \forall x\left( {P\left( x \right)} \right)$$ $${...
GATE CSE 2009
Which one of the following is the most appropriate logical formula to represent the statement: "$$Gold\,and\,silver\,orn...
GATE CSE 2009
Which of the following is the negation of $$$\left[ {\forall x,\alpha \to \left( {\exists y,\beta \to \left( {\forall...
GATE CSE 2008
$$P$$ and $$Q$$ are two propositions. Which of the following logical expressions are equivalent? $${\rm I}.$$ $${\rm P}\...
GATE CSE 2008
Which of the following first order formulae is logically valid? Here $$\alpha \left( x \right)$$ is a first order formul...
GATE CSE 2008
Let fsa and $$pda$$ be two predicates such that fsa$$(x)$$ means $$x$$ is a finite state automation, and pda$$(y)$$ mean...
GATE CSE 2008
If $$P$$, $$Q$$, $$R$$ are Boolean variables, then $$(P + \bar{Q}) (P.\bar{Q} + P.R) (\bar{P}.\bar{R} + \bar{Q})$...
GATE CSE 2008
Which one of these first-order logic formulae is valid?
GATE CSE 2007
Let $$P, Q$$, and $$R$$ be sets. Let $$\Delta $$ denote the symmetric difference operator defined as $$P\Delta Q = \left...
GATE CSE 2006
Consider the following first order logic formula in which $$R$$ is a binary relation symbol. $$\forall x\forall y\left( ...
GATE CSE 2006
A logical binary relation $$ \odot $$, is defined as follows: Let ~ be the unary negation (NOT) operator, with higher ...
GATE CSE 2006
Consider the following propositional statements: $${\rm P}1:\,\,\left( {\left( {A \wedge B} \right) \to C} \right) \equ...
GATE CSE 2006
Which one of the first order predicate calculus statements given below correctly expresses the following English stateme...
GATE CSE 2006
Let $$P(x)$$ and $$Q(x)$$ be arbitrary predicates. Which of the following statement is always TRUE?
GATE CSE 2005
Let $$P, Q$$ and $$R$$ be three atomic prepositional assertions. Let $$X$$ denotes $$\left( {P \vee Q} \right) \to R$$ a...
GATE CSE 2005
What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some studen...
GATE CSE 2005
Let $$p, q, r$$ and $$s$$ be four primitive statements. Consider the following arguments: $$P:\left[ {\left( {\neg p \ve...
GATE CSE 2004
The following propositional statement is $$$\left( {P \to \left( {Q \vee R} \right)} \right) \to \left( {\left( {P \wedg...
GATE CSE 2004
The following resolution rule is used in logic programming. Derive clause $$\left( {P \vee Q} \right)$$ from clauses $$\...
GATE CSE 2003
Let $$a, b, c, d$$ be propositions. Assume that the equivalences $$a \leftrightarrow \left( {b \vee \neg b} \right)$$ an...
GATE CSE 2000
Which one of the following is false? Read $$ \wedge $$ as AND, $$ \vee $$ as OR, $$ \sim $$ as NOT, $$ \to $$ as one way...
GATE CSE 1996
If the proposition $$\neg p \Rightarrow q$$ is true, then the truth value of the proposition $$\neg p \vee \left( {p \Ri...
GATE CSE 1995
Let $$p$$ and $$q$$ be propositions. Using only the truth table decide whether $$p \Leftrightarrow q$$ does not imply $...
GATE CSE 1994
Indicate which of the following well-formed formula are valid:
GATE CSE 1990

Marks 5

More
Determine whether each of the following is a tautology, a contradiction, or neither ("$$ \vee $$" is disjunction, "$$ \w...
GATE CSE 2002
(a) Show that the formula $$\left[ {\left( { \sim p \vee Q} \right) \Rightarrow \left( {q \Rightarrow p} \right)} \right...
GATE CSE 1999
Let $$\left( {\left\{ {p,\,q} \right\},\, * } \right)$$ be a semi group where $$p * p = q$$. Show that: (a) $$p * q ...
GATE CSE 1999
Show that proposition $$C$$ is a logical consequence of the formula $$A \wedge \left( {A \to \left( {B \vee C} \right) ...
GATE CSE 1993
Uses Modus ponens $$\left( {A,\,\,A \to B\,|\,\, = B} \right)$$ or resolution to show that the following set is inconsis...
GATE CSE 1992

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