Recent questions and answers in Mathematical Logic / GATE Overflow for GATE CSE

35 votes

4 answers

1

GO Classes CS 2025 | Weekly Quiz 2 | Propositional Logic | Question: 12
Two compound propositions are logically equivalent if they have the same truth table. For example, the following two compound propositions are logically equivalent: $\mathrm{p} \rightarrow \mathrm{q}$ ... propositional variables, how many compound propositions are there that are Not logically equivalent to each other?

Two compound propositions are logically equivalent if they have the same truth table.For example, the following two compound propositions are logically equivalent: $\math...

AjithAddala

1.2k views

AjithAddala answered 3 days ago

0 votes

1 answer

2

Linear Algebra AX=B
Consider a matrix A (n×m) ,X(m×n) and B(n×n) such that AX=B . If A has k linearly independent columns then what conclusions can we nake about the number of linearly independent columns of B.

Consider a matrix A (n×m) ,X(m×n) and B(n×n) such that AX=B . If A has k linearly independent columns then what conclusions can we nake about the number of linearly in...

Sahil5635

78 views

Sahil5635 answered 4 days ago

3 votes

2 answers

3

Poset
Consider the poset ({3,5,9,15,24,45},|). Which of the following is correct for the given poset? A. There exists a least element but not a greatest element B. There exists a greatest element but not a least element C. There exists a greatest element and a least element D. There does not exist a greatest element and a least element

Consider the poset ({3,5,9,15,24,45},|). Which of the following is correct for the given poset? A. There exists a least element but not a greatest elementB. There exists ...

Sahil5635

135 views

Sahil5635 answered 4 days ago

0 votes

1 answer

4

self doubt
how to check the validity of an a argument using laws of logics

how to check the validity of an a argument using laws of logics

Sahil5635

44 views

Sahil5635 answered Apr 15

3 votes

2 answers

5

self doubt
Twin primes are pairs of numbers p and p+2 such that both are primes-for instance, 5 and 7, 11 and 13, 41 and 43. The Twin Prime Conjecture says that there are infinitely many twin primes. Let TwinPrime(n) be a predicate that is true if n and n+2 are twin primes. Which of the ... (n)) ∃m. ∀n. n ≤ m implies TwinPrime(n) ∃m. ∀n. TwinPrime(n) implies n ≤ m ∀m. ∃n. n ≤ m and TwinPrime(n)

Twin primes are pairs of numbers p and p+2 such that both are primes—for instance, 5 and 7, 11 and 13, 41 and 43. The Twin Prime Conjecture says that there are infinite...

Deepak Poonia

589 views

Deepak Poonia answered Apr 14

26 votes

6 answers

6

GO Classes CS 2025 | Weekly Quiz 1 | Propositional Logic | Question: 15
Consider the following popular puzzle. A boy and a girl are talking. One of them has black hair, another has white hair. I am a boy said the child with black hair. I am a girl said the child with white hair ... Which of them is lying? The boy only The girl only Both of them Information is not sufficient to find out the liar

Consider the following popular puzzle.A boy and a girl are talking. One of them has black hair, another has white hair.“I am a boy” said the child with black hair.“...

KG

1.6k views

KG answered Apr 14

58 votes

8 answers

7

GO Classes CS 2025 | Weekly Quiz 1 | Propositional Logic | Question: 14
Consider the following popular puzzle. When asked for the ages of her three children, Mrs. Baker says that Alice is her youngest child if Bill is not her youngest child, and that Alice is not her youngest child ... is her youngest child. Carl is her youngest child. Information is not sufficient to find out the youngest child.

Consider the following popular puzzle.When asked for the ages of her three children, Mrs. Baker says that “Alice is her youngest child if Bill is not her youngest child...

KG

3.7k views

KG answered Apr 14

31 votes

5 answers

8

GATE CSE 1989 | Question: 14a
Symbolize the expression "Every mother loves her children" in predicate logic.

Symbolize the expression "Every mother loves her children" in predicate logic.

Ayon Sinha

6.0k views

Ayon Sinha answered Apr 12

1 votes

2 answers

9

Does Either...Or means Exclusive Or or Inclusive Or?
Let's take a compound propositions Either it is below freezing or it is snowing. Now if $p$: it is below freezing $q$: it is snowing Will it be $p \vee q$ or $p \oplus q$? There are some instances where semantics are ... both cases can't be true, because if you are ill you can't appear for example and you must be in one state.

Let's take a compound propositionsEither it is below freezing or it is snowing.Now if$p$: it is below freezing $q$: it is snowingWill it be $p \vee q$ or $p \oplus q$? Th...

dhruvak23

181 views

dhruvak23 answered Apr 11

4 votes

2 answers

10

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 6
The implies connective $\rightarrow$ is one of the stranger connectives in propositional logic. Below are a series of statements regarding implications. Which of the following statements is/are TRUE? For any propositions $P$ ... $R,$ the following statement is always true: $(P \rightarrow Q) \vee (R \rightarrow Q)$.

The “implies” connective “$\rightarrow$” is one of the stranger connectives in propositional logic. Below are a series of statements regarding implications.Which ...

ayushgemini

356 views

ayushgemini answered Apr 9

1 votes

1 answer

11

GATE 2022 | Linear Algebra | Question 11
Let 𝑴 be a 2 × 2 real matrix such that $(𝑰 + 𝑴)^-1 = 𝑰 − 𝛼𝑴$, where 𝛼 is a non-zero real number and 𝑰 is the 2 × 2 identity matrix. If the trace of the matrix 𝑴 is 3, then the value of 𝛼 is (A) 3/4 (B) 1/3 (C) 1/2 (D) 1/4

Let 𝑴 be a 2 × 2 real matrix such that $(𝑰 + 𝑴)^-1 = 𝑰 − 𝛼𝑴$, where 𝛼 is a non-zero real number and 𝑰 is the 2 × 2 identity matrix. If the tra...

C1h2a3i4t5

117 views

C1h2a3i4t5 answered Apr 8

0 votes

1 answer

12

BME entrance exam
A hash function h maps 16-bit inputs to 8-bit hash values. What is the largest k such that in any set of 1000 inputs, there are at least k inputs that h maps to the same hash value?

A hash function h maps 16-bit inputs to 8-bit hash values. What is the largest k such that in any set of 1000 inputs, there are at least k inputs that h maps to the same ...

bhanu7486

66 views

bhanu7486 answered Apr 8

0 votes

1 answer

13

Made Easy Mock Test 2

Sherry_22

273 views

Sherry_22 answered Apr 8

0 votes

1 answer

14

Internet
A non empty set A is termed as an algebraic structure ________ a)with respect to binary operation * b)with respect to ternary operation ? c)with respect to binary operation + d)with respect to unary operation –

A non empty set A is termed as an algebraic structure ________a)with respect to binary operation *b)with respect to ternary operation ?c)with respect to binary operation ...

bhanu7486

86 views

bhanu7486 answered Apr 6

2 votes

2 answers

15

Memory Based GATE DA 2024 | Question: 33
Which of the following are tautologies? $x \land \neg y \Rightarrow y \rightarrow x$ $\neg x \land y \Rightarrow \neg x \rightarrow y$ $x \land \neg y \Rightarrow \neg x \rightarrow y$ $\neg x \land y \Rightarrow y \rightarrow x$

Which of the following are tautologies? $x \land \neg y \Rightarrow y \rightarrow x$$\neg x \land y \Rightarrow \neg x \rightarrow y$\(x \land \neg y \Rightarrow \ne...

i_m_sudip

256 views

i_m_sudip answered Apr 3

0 votes

1 answer

16

Permutation and combination
Your mother-in-law buys 1000 small gifts to give to relatives for Christmas. Each of the 1000 things in different. There are 25 relatives to give gifts to. How many ways are there to distribute the gifts? The correct answer is $25^{1000}$. I ... ? I know some people may feel its silly question but please trust me many people like me are confused with this doubt.

Your mother-in-law buys 1000 small gifts to give to relatives for Christmas. Each of the 1000 things in different. There are 25 relatives to give gifts to. How many ways ...

anonymous1

106 views

anonymous1 answered Apr 2

0 votes

0 answers

17

BME entrance exam
A data file of 1,000,000 records must be stored using bucket hashing. Record length is 350 bytes, the net capacity of a block is 2000 bytes. Keys occupy 32 bytes, while pointers need 8 bytes. Each bucket consists of 2 blocks. How many blocks are needed at least to store the whole structure (i.e. data file + hash table) on disc? Select one:

A data file of 1,000,000 records must be stored using bucket hashing. Record length is 350 bytes, the net capacity of a block is 2000 bytes. Keys occupy 32 bytes, while p...

Ilqar

69 views

Ilqar asked Apr 2

12 votes

3 answers

18

GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 58
Let $\mathrm{F}$ and $\mathrm{G}$ be two propositional formulae. Which of the following is/are True? If $F \vee G$ is a tautology then at least one of $F, G$ is a tautology. If $F \wedge G$ is a contradiction then at ... $G$ is a tautology. If $F \rightarrow G$ is a contradiction then $F$ is a tautology and $G$ is a contradiction.

Let $\mathrm{F}$ and $\mathrm{G}$ be two propositional formulae.Which of the following is/are True?If $F \vee G$ is a tautology then at least one of $F, G$ is a tautology...

Lakshmi Narayana404

853 views

Lakshmi Narayana404 answered Apr 2

75 votes

8 answers

19

GATE CSE 2016 Set 1 | Question: 1
Let $p, q, r, s$ represents the following propositions. $p:x\in\left\{8, 9, 10, 11, 12\right\}$ $q:$ $x$ is a composite number. $r:$ $x$ is a perfect square. $s:$ $x$ is a prime number. The integer $x\geq2$ which satisfies $\neg\left(\left(p\Rightarrow q\right) \wedge \left(\neg r \vee \neg s\right)\right)$ is ____________.

Let $p, q, r, s$ represents the following propositions.$p:x\in\left\{8, 9, 10, 11, 12\right\}$$q:$ $x$ is a composite number.$r:$ $x$ is a perfect square.$s:$ $x$ is a pr...

i_m_sudip

13.1k views

i_m_sudip answered Mar 30

12 votes

2 answers

20

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 8
The Logic Problem, taken from $\text{"WFF'N PROOF, The Game of Logic"}$ has these two assumptions: "Logic is difficult or not many students like logic." "If mathematics is easy, then logic ... is not easy or logic is difficult." "Logic is not difficult or mathematics is not easy."

The Logic Problem, taken from $\text{"WFF'N PROOF, The Game of Logic"}$ has these two assumptions:"Logic is difficult or not many students like logic.""If mathematics is ...

tejashmore25

452 views

tejashmore25 answered Mar 29

7 votes

2 answers

21

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 10
Let $F$ and $G$ be two propositional formula. Which of the following is/are True? $F \vee G$ is a tautology iff at least one of them is a tautology If $F \rightarrow G$ is a tautology and $F$ ... is a tautology. $(F \rightarrow G) \wedge (F \rightarrow \neg G)$ is a tautology iff $F$ is a contradiction.

Let $F$ and $G$ be two propositional formula.Which of the following is/are True?$F \vee G$ is a tautology iff at least one of them is a tautologyIf $F \rightarrow G$ is a...

samitmohan

734 views

samitmohan answered Mar 29

10 votes

6 answers

22

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 3
Consider the following atomic propositions: $\text{R}$: It is Raining $\text{S}$ ... , and vice versa It is raining is equivalent to sonu is sick It is raining or sonu is sick but not both

Consider the following atomic propositions:$\text{R}$: It is Raining$\text{S}$: Sonu is SickWhich of the following is/are correct English Translation of the following log...

samitmohan

684 views

samitmohan answered Mar 29

6 votes

2 answers

23

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 5
Which of the following compound propositions is/are a tautology? $[p \land (p \rightarrow q)] \rightarrow q$ $[q \land (p \rightarrow q)] \rightarrow p$ $[(p \lor q) \land (r \rightarrow p) \land (r \rightarrow q)] \rightarrow r$ $[(p \lor q) \land (p \rightarrow r) \land (q \rightarrow r)] \rightarrow r$

Which of the following compound propositions is/are a tautology?$[p \land (p \rightarrow q)] \rightarrow q$$[q \land (p \rightarrow q)] \rightarrow p$$[(p \lor q) \land (...

samitmohan

129 views

samitmohan answered Mar 29

7 votes

2 answers

24

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 4
Consider the following arguments. $\text{Argument 1:}$ ... ? Only Argument $1$ is valid. Only Argument $2$ is valid. Both Arguments are valid. No Argument is valid.

Consider the following arguments.$\text{Argument 1:}$ Kerry errs or Myrna fails to show. If Kerry errs, then he does not break the record. Myrna fails to show. Therefore,...

https_guru

458 views

https_guru answered Mar 29

21 votes

6 answers

25

GO Classes CS 2025 | Weekly Quiz 3 | Propositional Logic | Question: 9
Which of the following statements is/are true? The argument form with premises $p_1, p_2, \ldots, p_n$ and conclusion $q \rightarrow r$ is valid iff the argument form with premises $p_1, p_2, \ldots, p_n, r$ ... the argument form with premises $p_1, p_2, \ldots, p_n, \sim r$, and conclusion $\sim q$ is valid.

Which of the following statements is/are true?The argument form with premises $p_1, p_2, \ldots, p_n$ and conclusion $q \rightarrow r$ is valid iff the argument form with...

Lakshmi Narayana404

917 views

Lakshmi Narayana404 answered Mar 29

7 votes

4 answers

26

GATE CSE 2024 | Set 2 | Question: 2
Let $p$ and $q$ be the following propositions: $p$ : Fail grade can be given. $q$ : Student scores more than $50 \%$ marks. Consider the statement: "Fail grade cannot be given when student scores more than $50 \%$ marks." ... above statement in propositional logic? $q \rightarrow \neg p$ $q \rightarrow p$ $p \rightarrow q$ $\neg p \rightarrow q$

Let $p$ and $q$ be the following propositions:$p$ : Fail grade can be given.$q$ : Student scores more than $50 \%$ marks.Consider the statement: "Fail grade c...

Bhaskar_Saini

3.4k views

Bhaskar_Saini answered Mar 29

14 votes

3 answers

27

GO Classes CS 2025 | Weekly Quiz 1 | Propositional Logic | Question: 10
Given the truth table of a Binary Operation \$ as follows: $ ... }$ Identify the matching Boolean Expression. $X \$ \neg Y$ $\neg X \$ Y$ $\neg X \$ \neg Y$ none of the options

Given the truth table of a Binary Operation \$ as follows:$$\begin{array}{|l|l|l|l|} \hline {} \text{X} & \text{Y }& \text{X\$Y }\\ \hline \text{1} & \text{0 }& ...

Ayon Sinha

1.1k views

Ayon Sinha answered Mar 28

36 votes

3 answers

28

GO Classes CS 2025 | Weekly Quiz 2 | Propositional Logic | Question: 8
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi$ $\Rightarrow$ $\psi$ . A $\textit{Craig interpolant}$ of $\varphi$ and $\psi$ ... interpolant for $\varphi$ and $\psi?$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$

Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi$ $\Rightarrow$...

Sahil5635

781 views

Sahil5635 answered Mar 27

9 votes

5 answers

29

GO Classes CS 2025 | Weekly Quiz 1 | Propositional Logic | Question: 2
A sufficient condition for a triangle $T$ be a right triangle is that $a^2+b^2=c^2$. An equivalent statement is If $T$ is a right triangle then $a^2+b^2=c^2$. If $a^2+b^2=c^2$ then $T$ is a right triangle. If $a^2+b^2\neq c^2$ then $T$ is not a right triangle. $T$ is a right triangle only if $a^2+b^2=c^2$.

A sufficient condition for a triangle $T$ be a right triangle is that $a^2+b^2=c^2$. An equivalent statement isIf $T$ is a right triangle then $a^2+b^2=c^2$.If $a^2+b^2=c...

tejashmore25

1.5k views

tejashmore25 answered Mar 27

73 votes

8 answers

30

GATE CSE 2017 Set 1 | Question: 02
Consider the first-order logic sentence $F:\forall x(\exists yR(x,y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$? $\exists y(\exists xR(x,y))$ $\exists y(\forall xR(x,y))$ $\forall y(\exists xR(x,y))$ $¬\exists x(\forall y¬R(x,y))$ IV only I and IV only II only II and III only

Consider the first-order logic sentence $F:\forall x(\exists yR(x,y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$?$\exists y(\ex...

TusharRana

17.4k views

TusharRana answered Mar 26

42 votes

9 answers

31

GATE CSE 2020 | Question: 39
Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$. $\forall x (p(x) \vee W) \equiv \forall x \: ( px) \vee W$ ... $\exists x(p(x) \rightarrow W) \equiv \forall x \: p(x) \rightarrow W$

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) \equ...

TusharRana

17.3k views

TusharRana answered Mar 25

63 votes

7 answers

32

GATE IT 2006 | Question: 21
Consider the following first order logic formula in which $R$ is a binary relation symbol. $∀x∀y (R(x, y) \implies R(y, x))$ The formula is satisfiable and valid satisfiable and so is its negation unsatisfiable but its negation is valid satisfiable but its negation is unsatisfiable

Consider the following first order logic formula in which $R$ is a binary relation symbol.$∀x∀y (R(x, y) \implies R(y, x))$The formula issatisfiable and validsatisfia...

TusharRana

13.5k views

TusharRana answered Mar 25

7 votes

2 answers

33

TIFR CSE 2019 | Part B | Question: 4
Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi \Rightarrow \psi$ . A $\textit{Craig interpolant}$ of $\varphi$ and $\psi$ is a propositional formula $\mu$ ... interpolant for $\varphi$ and $\psi$ ? $q$ $\varphi$ itself $q \vee s$ $q \vee r$ $\neg q \wedge s$

Let $\varphi$ be a propositional formula on a set of variables $A$ and $\psi$ be a propositional formula on a set of variables $B$ , such that $\varphi \Rightarrow \p...

Sahil5635

1.4k views

Sahil5635 answered Mar 25

25 votes

4 answers

34

TIFR CSE 2010 | Part A | Question: 8
Which of the following is NOT necessarily true? { Notation: The symbol ''$\neg$''notes negation; $P (x, y)$ means that for given $x$ and $y$, the property $P(x, y)$ is true }. $(∀x∀y P(x, y)) \Rightarrow (∀y∀x P(x, y))$ ... $(∃x∀y P(x, y)) \Rightarrow (∀y∃x P(x, y))$ $(∀x∃y P(x, y)) \Rightarrow (∃y∀x P(x, y))$

Which of the following is NOT necessarily true? { Notation: The symbol ''$\neg$''notes negation; $P (x, y)$ means that for given $x$ and $y$, the property $P(x, y)$ is tr...

Sahil5635

3.3k views

Sahil5635 answered Mar 24

0 votes

1 answer

35

Discrete mathematics Ch 1 : Propositional logic , Topic 2 : Logical operators or connectives
Which of the following is the negation of x is even iff x is divisible by 2 a) (x is even or x is not divisible by 2) and (x is not even or x is divisible by 2) b) (x is even and x is not divisible by 2) ... is divisible by 2) c) x is not even iff x is not divisible by 2 d) x is even if x is divisible by 2

Which of the following is the negation of x is even iff x is divisible by 2a) (x is even or x is not divisible by 2) and (x is not even or x is divisible by 2)b) (x is ev...

KraVLonE

149 views

KraVLonE answered Mar 20

4 votes

4 answers

36

Self Doubt: Mathematical Logic
Is the assertion "This statement is false" a proposition?

Is the assertion "This statement is false" a proposition?

TusharRana

2.2k views

TusharRana answered Mar 18

7 votes

3 answers

37

GO Classes CS 2025 | Weekly Quiz 2 | Propositional Logic | Question: 2
Which of the following expressions is false? $p \rightarrow q \equiv q \rightarrow p$ $\neg(p \vee q) \equiv \neg p \wedge \neg q$ $p \rightarrow q \equiv \neg q \rightarrow \neg p$ none of the above

Which of the following expressions is false?$p \rightarrow q \equiv q \rightarrow p$$\neg(p \vee q) \equiv \neg p \wedge \neg q$$p \rightarrow q \equiv \neg q \rightarrow...

i_m_sudip

363 views

i_m_sudip answered Mar 15

5 votes

2 answers

38

GO Classes CS 2025 | Weekly Quiz 2 | Propositional Logic | Question: 13
Let $p,q,r$ be three propositional variables. Which of the following statements is/are false? $p \rightarrow(q \vee r)) \equiv((p \wedge \neg q) \rightarrow r)$ $(p \wedge q) \vee r \equiv p \wedge(q \vee r)$ ... is FALSE then $(q \rightarrow p)$ is TRUE. If $(p \rightarrow q)$ is TRUE then $(q \rightarrow p)$ is FALSE.

Let $p,q,r$ be three propositional variables.Which of the following statements is/are false?$p \rightarrow(q \vee r)) \equiv((p \wedge \neg q) \rightarrow r)$$(p \wedge q...

i_m_sudip

430 views

i_m_sudip answered Mar 15

35 votes

4 answers

39

GATE CSE 2005 | Question: 40
Let $P, Q,$ and $R$ be three atomic propositional assertions. Let $X$ denote $( P ∨ Q ) → R$ and $Y$ denote $(P → R) ∨ (Q → R).$ Which one of the following is a tautology? $X ≡ Y$ $X → Y$ $Y → X$ $¬Y → X$

Let $P, Q,$ and $R$ be three atomic propositional assertions. Let $X$ denote $( P ∨ Q ) → R$ and $Y$ denote $(P → R) ∨ (Q → R).$ Which one of the following is a...

i_m_sudip

6.6k views

i_m_sudip answered Mar 13

14 votes

8 answers

40

GATE CSE 2021 Set 1 | Question: 7
Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $S_1: (\neg p\wedge(p\vee q))\rightarrow q$ $S_2: q\rightarrow(\neg p\wedge(p\vee q))$ Which one of the following choices is correct? Both $S_1$ and ... but $S_2$ is not a tautology $S_1$ is not a tautology but $S_2$ is a tautology Neither $S_1$ nor $S_2$ is a tautology

Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic.$S_1: (\neg p\wedge(p\vee q))\rightarrow q$$S_2: q\rightarrow(\neg p\wedge...

ritiksri8

8.4k views

ritiksri8 answered Mar 9

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