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GATE CSE
Mathematical Logic
Discrete Mathematics
Previous Years Questions
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Marks 1
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Consider the following expressions: $$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(i)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ false $$\,...
GATE CSE 2016 Set 2
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Let $$p,q,r,s$$ represent the following propositions. $$p:\,\,\,x \in \left\{ {8,9,10,11,12} \right\}$$ $$q:\,\,\,x$$ is...
GATE CSE 2016 Set 1
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In a room there are only two types of people, namely Type $$1$$ and Type $$2.$$ Type $$1$$ people always tell the truth ...
GATE CSE 2015 Set 3
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Consider the following two statements. $$S1:$$ If a candidate is known to be corrupt, then he will not be elected $$S2:$...
GATE CSE 2015 Set 2
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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
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Consider the statement "Not all that glitters is gold" Predicate glitters$$(x)$$ is true if $$x$$ glitters and predic...
GATE CSE 2014 Set 1
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The truth table Represents the Boolean function...
GATE CSE 2012
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What is the correct translation of the following statement into mathematical logic? "Some real numbers are rational"
GATE CSE 2012
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Consider the following logical inferences. $${{\rm I}_1}:$$ If it rains then the cricket match will not be played. The...
GATE CSE 2012
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A set of Boolean connectives is functionally complete if all Boolean function can be synthesized using those, Which of t...
GATE CSE 2008
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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
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Identify the correct translation into logical notation of the following assertion. $$Some\,boys\,in\,the\,class\,are\,t...
GATE CSE 2004
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"If X then Y unless Z" is represented by which of the following formulas in propositional logic? (" $$\neg $$ " is negat...
GATE CSE 2002
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Consider two well-formed formulas in propositional logic $$F1:P \Rightarrow \neg P$$ $$F2:\left( {P \Rightarrow \neg P}...
GATE CSE 2001
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What is the converse of the following assertion? I stay only if you go
GATE CSE 1998
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The proposition $$p \wedge \left( { \sim p \vee q} \right)$$ is
GATE CSE 1993
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Which of the following predicate calculus statements is/are valid?
GATE CSE 1992
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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
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Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any fre...
GATE CSE 2020
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Consider the first-order logic sentence $$\varphi \equiv \,\,\,\,\,\,\,\exists s\exists t\exists u\forall v\forall w$$...
GATE CSE 2018
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Which one of the following well-formed formulae in predicate calculus is NOT valid?
GATE CSE 2016 Set 2
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Which one of the following well formed formulae is a tautology?
GATE CSE 2015 Set 2
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Which one of the following Boolean expressions is NOT A tautology?
GATE CSE 2014 Set 2
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The CORRECT formula for the sentence, "not all rainy days are cold" is
GATE CSE 2014 Set 3
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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
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Which one of the following is NOT logically equivalent to $$\neg \exists x\left( {\forall y\left( \alpha \right) \wedge...
GATE CSE 2013
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What is the logical translation of the following statement? "None of my friends are perfect."
GATE CSE 2013
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Which one of the following options is correct given three positive integers $$x, y$$ and $$z$$, and a predicate $$P\lef...
GATE CSE 2011
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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
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The binary operation ◻ is defined as follows: Which one of the following is equivalence to $$P \vee Q$$? ...
GATE CSE 2009
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Consider the following well-formed formulae: $${\rm I}.$$ $$\,\,\neg \forall x\left( {P\left( x \right)} \right)$$ $${...
GATE CSE 2009
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Which one of the following is the most appropriate logical formula to represent the statement: "$$Gold\,and\,silver\,orn...
GATE CSE 2009
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Which of the following is the negation of $$$\left[ {\forall x,\alpha \to \left( {\exists y,\beta \to \left( {\forall...
GATE CSE 2008
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$$P$$ and $$Q$$ are two propositions. Which of the following logical expressions are equivalent? $${\rm I}.$$ $${\rm P}\...
GATE CSE 2008
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Which of the following first order formulae is logically valid? Here $$\alpha \left( x \right)$$ is a first order formul...
GATE CSE 2008
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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
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If $$P$$, $$Q$$, $$R$$ are Boolean variables, then $$(P + \bar{Q}) (P.\bar{Q} + P.R) (\bar{P}.\bar{R} + \bar{Q})$...
GATE CSE 2008
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Which one of these first-order logic formulae is valid?
GATE CSE 2007
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Let $$P, Q$$, and $$R$$ be sets. Let $$\Delta $$ denote the symmetric difference operator defined as $$P\Delta Q = \left...
GATE CSE 2006
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Consider the following first order logic formula in which $$R$$ is a binary relation symbol. $$\forall x\forall y\left( ...
GATE CSE 2006
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A logical binary relation $$ \odot $$, is defined as follows: Let ~ be the unary negation (NOT) operator, with higher ...
GATE CSE 2006
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Consider the following propositional statements: $${\rm P}1:\,\,\left( {\left( {A \wedge B} \right) \to C} \right) \equ...
GATE CSE 2006
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Which one of the first order predicate calculus statements given below correctly expresses the following English stateme...
GATE CSE 2006
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Let $$P(x)$$ and $$Q(x)$$ be arbitrary predicates. Which of the following statement is always TRUE?
GATE CSE 2005
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Let $$P, Q$$ and $$R$$ be three atomic prepositional assertions. Let $$X$$ denotes $$\left( {P \vee Q} \right) \to R$$ a...
GATE CSE 2005
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What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some studen...
GATE CSE 2005
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Let $$p, q, r$$ and $$s$$ be four primitive statements. Consider the following arguments: $$P:\left[ {\left( {\neg p \ve...
GATE CSE 2004
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The following propositional statement is $$$\left( {P \to \left( {Q \vee R} \right)} \right) \to \left( {\left( {P \wedg...
GATE CSE 2004
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The following resolution rule is used in logic programming. Derive clause $$\left( {P \vee Q} \right)$$ from clauses $$\...
GATE CSE 2003
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Let $$a, b, c, d$$ be propositions. Assume that the equivalences $$a \leftrightarrow \left( {b \vee \neg b} \right)$$ an...
GATE CSE 2000
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Which one of the following is false? Read $$ \wedge $$ as AND, $$ \vee $$ as OR, $$ \sim $$ as NOT, $$ \to $$ as one way...
GATE CSE 1996
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If the proposition $$\neg p \Rightarrow q$$ is true, then the truth value of the proposition $$\neg p \vee \left( {p \Ri...
GATE CSE 1995
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Let $$p$$ and $$q$$ be propositions. Using only the truth table decide whether $$p \Leftrightarrow q$$ does not imply $...
GATE CSE 1994
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Indicate which of the following well-formed formula are valid:
GATE CSE 1990
GO TO QUESTION
Marks 5
More
Determine whether each of the following is a tautology, a contradiction, or neither ("$$ \vee $$" is disjunction, "$$ \w...
GATE CSE 2002
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(a) Show that the formula $$\left[ {\left( { \sim p \vee Q} \right) \Rightarrow \left( {q \Rightarrow p} \right)} \right...
GATE CSE 1999
GO TO QUESTION
Let $$\left( {\left\{ {p,\,q} \right\},\, * } \right)$$ be a semi group where $$p * p = q$$. Show that: (a) $$p * q ...
GATE CSE 1999
GO TO QUESTION
Show that proposition $$C$$ is a logical consequence of the formula $$A \wedge \left( {A \to \left( {B \vee C} \right) ...
GATE CSE 1993
GO TO QUESTION
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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