Propositional Logic


Question 1
Geetha has a conjecture about integers, which is of the form
\forall x\left [P(x)\Rightarrow \exists yQ(x,y) \right ]
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?
A
\exists x\left [P(x)\wedge \forall yQ(x,y) \right ]
B
\forall x \forall y Q(x,y)
C
\exists y \forall x \left [P(x) \Rightarrow Q(x,y) \right ]
D
\exists x \left [P(x) \wedge \exists y Q(x,y) \right ]
GATE CSE 2023   Discrete Mathematics
Question 2
The number of arrangements of six identical balls in three identical bins is ____.
A
7
B
8
C
12
D
5
GATE CSE 2022   Discrete Mathematics


Question 3
Choose the correct choice(s) regarding the following proportional logic assertion S:

S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))
[MSQ]
A
S is neither a tautology nor a contradiction
B
S is a tautology
C
S is a contradiction
D
The antecedent of S is logically equivalent to the consequent of S
GATE CSE 2021 SET-2   Discrete Mathematics
Question 4
Consider the two statements.

S1: There exist random variables X and Y such that \left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2 > \textsf{Var}[X]\textsf{Var}[Y]
S2: For all random variables X and Y, \textsf{Cov}[X,Y]=\mathbb E \left[|X-\mathbb E[X]||Y-\mathbb E[Y]|\right ]

Which one of the following choices is correct?
A
Both S1 and S2 are true.
B
S1 is true, but S2 is false.
C
S1 is false, but S2 is true.
D
Both S1 and S2 are false.
GATE CSE 2021 SET-1   Discrete Mathematics
Question 5
Let p and q be two propositions. Consider the following two formulae in propositional logic.

S1: (\neg p\wedge(p\vee q))\rightarrow q
S2: q\rightarrow(\neg p\wedge(p\vee q))

Which one of the following choices is correct?
A
Both S1 and S2 are tautologies.
B
S1 is a tautology but S2 is not a tautology
C
S1 is not a tautology but S2 is a tautology
D
Niether S1 nor S2 is a tautology
GATE CSE 2021 SET-1   Discrete Mathematics




There are 5 questions to complete.

17 thoughts on “Propositional Logic”

  1. Qno. 40. Correction in option 4
    The actual option is ∀x [(tiger(x) ∨ lion(x)) → (hungry(x) ∨ threatened(x)) → attacks(x)]
    At the place of “∧ , there will be ” ∨”.

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  2. In the question 23 please update the answer. It is not 0, instead it needs to be ∀x(∃y(¬α)→∃z(¬β))

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  3. In the question 29 please update the question. In the option iv it is not ¬∃x(¬P(x)), instead it needs to be ∃x(¬P(x))

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