Recent questions tagged mathematical-logic / GATE Overflow for GATE CSE

3 votes

2 answers

301

GO Classes 2023 | Weekly Quiz 7 | Question: 4
Consider the following statement $\text{S}$ in an universe $\text{U}.$ $\text{S} : \forall x \forall y (x = y)$ What is the maximum cardinality of $\text{U}$ such that $\text{S}$ is true?

Consider the following statement $\text{S}$ in an universe $\text{U}.$$\text{S} : \forall x \forall y (x = y)$What is the maximum cardinality of $\text{U}$ such that $\te...

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376 views

GO Classes asked Apr 14, 2022

1 votes

1 answer

302

GO Classes 2023 | Weekly Quiz 7 | Question: 6
Consider the following logical inferences : $\text{S1} :$ If I study Discrete Mathematics, then I will study Computer Science. If I study C, then I will study Algorithms. Therefore, If I study Discrete Mathematics or C then I will study ... correct but $\text{S2}$ is a correct inference Both $\text{S1}$ and $\text{S2}$ are not correct inferences

Consider the following logical inferences :$\text{S1} :$“If I study Discrete Mathematics, then I will study Computer Science.”“If I study C, then I will study Algor...

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425 views

GO Classes asked Apr 14, 2022

1 votes

1 answer

303

GO Classes 2023 | Weekly Quiz 7 | Question: 8
Consider the formula $\exists x \exists y \exists z(\text{R}(x, y) \wedge \text{R}(z, y) \wedge \text{R}(x, z) \wedge \neg \text{R}(z, x)).$ For which of the following interpretations, is this formula true? $(\text{N}$ ... $\text{R}(x,y) : y = x0 \;\text{or}\; y = x1.$

Consider the formula $\exists x \exists y \exists z(\text{R}(x, y) \wedge \text{R}(z, y) \wedge \text{R}(x, z) \wedge \neg \text{R}(z, x)).$For which of the following int...

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371 views

GO Classes asked Apr 14, 2022

5 votes

1 answer

304

GO Classes 2023 | Weekly Quiz 5 | Question: 1
Consider the following atomic propositions: $\text{R}$: It is Raining $\text{S}$ ... is raining, 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...

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708 views

GO Classes asked Mar 30, 2022

5 votes

1 answer

305

GO Classes 2023 | Weekly Quiz 5 | Question: 2
Let $\text{P}$ be a propositional variable. Which of the following propositions are tautologies? $\text{P}$ $\text{P} \Rightarrow \text{P}$ $(\text{P} \Rightarrow \text{P}) \Rightarrow \text{P}$ $\text{P} \Rightarrow(\text{P} \Rightarrow \text{P})$

Let $\text{P}$ be a propositional variable.Which of the following propositions are tautologies?$\text{P}$$\text{P} \Rightarrow \text{P}$$(\text{P} \Rightarrow \text{P}) \...

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422 views

GO Classes asked Mar 30, 2022

6 votes

1 answer

306

GO Classes 2023 | Weekly Quiz 5 | Question: 3
Which of the two following propositions are equivalent in the sense that one can always be substituted for the other one in any proposition without changing its truth value? first proposition $:\text{P} \Rightarrow \text{Q};$ ... $:\neg \text{P};$ second proposition $:\neg \text{P} \vee \text{Q}$

Which of the two following propositions are equivalent in the sense that one can always be substituted for the other one in anyproposition without changing its truth valu...

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591 views

GO Classes asked Mar 30, 2022

7 votes

1 answer

307

GO Classes 2023 | Weekly Quiz 5 | Question: 4
Let $\text{ KB }$ stand for $\text{ knowledge base }$ which is a set of premises/statements, as mentioned following: Let $\text{S}$ be a propositional formula. Which of the following is $\text{ possible }$? $(\text{KB}\models \text{S})$ ... $(\text{KB} \models \text{S})$ and $(\text{KB} \not \models \neg \text{S})$

Let $\text{“KB”}$ stand for $\text{“knowledge base”}$ which is a set of premises/statements, as mentioned following: Let $\text{S}$ be a propositional formula.Whi...

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512 views

GO Classes asked Mar 30, 2022

3 votes

1 answer

308

GO Classes 2023 | Weekly Quiz 5 | Question: 7
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, Kerry does break the record. $\text{Argument 2:}$ If Tasha leaves, ... is true? 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,...

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391 views

GO Classes asked Mar 30, 2022

3 votes

0 answers

309

GO Classes 2023 | Weekly Quiz 5 | Question: 8
Consider the following arguments. $\text{Argument 1:}$ If it does not rain or it is not foggy, then the sailing race will be held, and life-saving demonstrations will go on. If the sailing race is held, then the trophy will be awarded. ... is true? Only Argument $1$ is valid. Only Argument $2$ is valid. Both Arguments are valid. No Argument is valid.

Consider the following arguments.$\text{Argument 1:}$ If it does not rain or it is not foggy, then the sailing race will be held, and life-saving demonstrations will go o...

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671 views

GO Classes asked Mar 30, 2022

6 votes

1 answer

310

GO Classes 2023 | Weekly Quiz 5 | Question: 9
Consider the following puzzle. Five persons $\text{A, B, C, D, E}$ are in a compartment in a train. $\text{A, C, E}$ are men, and $\text{B, D}$ ... given above, $4$ are true, and $4$ are false. Who is the murderer (Assume exactly one person is the murderer)? A B C D

Consider the following puzzle.Five persons $\text{A, B, C, D, E}$ are in a compartment in a train. $\text{A, C, E}$ are men, and $\text{B, D}$ are women. The train passes...

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561 views

GO Classes asked Mar 30, 2022

4 votes

0 answers

311

GO Classes 2023 | Weekly Quiz 5 | Question: 10
Let $\alpha, \beta$ be two propositional formulas. Then which of the following is/are false? If $(\neg \alpha \vee \beta)$ is valid, then $\alpha \models \beta$. If $(\neg \alpha \vee \beta)$ is valid, then $\neg \alpha \models \beta$ ... $\alpha \models \neg \beta$. If $(\neg \alpha \vee \beta)$ is valid, then $\neg \beta \models \neg \alpha$.

Let $\alpha, \beta$ be two propositional formulas.Then which of the following is/are false?If $(\neg \alpha \vee \beta)$ is valid, then $\alpha \models \beta$.If $(\neg \...

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262 views

GO Classes asked Mar 30, 2022

4 votes

0 answers

312

GO Classes 2023 | Weekly Quiz 5 | Question: 11
Let $\alpha, \beta$ be two propositional formulas. Which of the following assertions is true? $\alpha$ is valid if and only if True $\models \alpha $. For any $\alpha$, False $\models \alpha$. $\alpha \models \beta $ if and only ... $\alpha \equiv \beta$ if and only if the sentence $(\alpha \Leftrightarrow \beta)$is valid.

Let $\alpha, \beta$ be two propositional formulas.Which of the following assertions is true?$\alpha$ is valid if and only if True $\models \alpha $.For any $\alpha$, Fals...

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268 views

GO Classes asked Mar 30, 2022

6 votes

1 answer

313

GO Classes 2023 | Weekly Quiz 5 | Question: 12
Let $\alpha, \beta $ be two propositional formulas. Which of the following assertions is true? $\alpha \models \beta$ if and only if the sentence $(\alpha \wedge \neg \beta)$ is unsatisfiable. If $\alpha\models \gamma $ ... $\alpha \models(\beta \vee \gamma)$ then $\alpha \models \beta $ or $\alpha \models \gamma $ (or both).

Let $\alpha, \beta $ be two propositional formulas.Which of the following assertions is true?$\alpha \models \beta$ if and only if the sentence $(\alpha \wedge \neg \beta...

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907 views

GO Classes asked Mar 30, 2022

5 votes

1 answer

314

GO Classes 2023 | Weekly Quiz 5 | Question: 13
Consider two formula : $\phi : p\wedge \neg q \rightarrow p\wedge q$ $\psi : \neg p$. Which of the following is/are true? $\phi$ is a logical consequence of the formula $\psi$ $\psi$ is a logical consequence of the formula $\phi$ $\phi$ is a logically equivalent to the formula $\psi$ Neither $\phi$, nor $\psi$, is a logical consequence of the other.

Consider two formula :$\phi : p\wedge \neg q \rightarrow p\wedge q$$\psi : \neg p$.Which of the following is/are true?$\phi$ is a logical consequence of the formula $\psi...

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404 views

GO Classes asked Mar 30, 2022

3 votes

1 answer

315

GO Classes 2023 | Weekly Quiz 5 | Question: 14
Which of the following logical consequences are correct? $(p \rightarrow q) |= \neg p \rightarrow \neg q$ $(p \rightarrow q) \wedge \neg q |= \neg p$ $p \rightarrow q \wedge r |= (p \rightarrow q) \rightarrow r$ $p \vee (\neg q \wedge r) |= q \vee \neg r \rightarrow p$

Which of the following logical consequences are correct?$(p \rightarrow q) |= \neg p \rightarrow \neg q$$(p \rightarrow q) \wedge \neg q |= \neg p$$p \rightarrow q \wedge...

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264 views

GO Classes asked Mar 30, 2022

3 votes

1 answer

316

GO Classes 2023 | Weekly Quiz 5 | Question: 15
Let's consider a propositional language where $p$ means Paola is happy , $q$ means Paola paints a picture , $r$ means Renzo is happy . Consider the following sentences and propositional logic formulas : $\text{X} :$ if Paola is happy and paints a picture, then Renzo isn't happy ... $\text{X-1, Y-2, Z-3}$ $\text{X-3, Y-2, Z-1}$

Let’s consider a propositional language where$p$ means “Paola is happy”,$q$ means “Paola paints a picture”,$r$ means “Renzo is happy”.Consider the following...

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801 views

GO Classes asked Mar 30, 2022

4 votes

1 answer

317

GO Classes 2023 | Weekly Quiz 3 | Question: 1
If $F_1, F_2$ and $F_3$ are propositional formulae/expressions, over same set of propositional variables, such that $F_1\wedge F_2\rightarrow F_3$ is a contradiction, then which of the following is/are necessarily true? Both $F_1$ and $F_2$ ... $F_1\wedge F_2$ is a tautology $F_3$ is a contradiction $F_1,\;F_2$ and $F_3$ all are contradictions.

If $F_1, F_2$ and $F_3$ are propositional formulae/expressions, over same set of propositional variables, such that $F_1\wedge F_2\rightarrow F_3$ is a contradiction, the...

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969 views

GO Classes asked Mar 23, 2022

4 votes

1 answer

318

GO Classes 2023 | Weekly Quiz 3 | Question: 2
If $F_1, F_2$ and $F_3$ are propositional formulae/expressions, over same set of propositional variables, such that $(F_1\rightarrow F_2)\rightarrow F_3$ is a contradiction, then which of the following is/are necessarily true? It is not ... Contradiction. $F_3$ is a contradiction. It is not possible that $F_1$ is Contingency and $F_2$ is Contradiction.

If $F_1, F_2$ and $F_3$ are propositional formulae/expressions, over same set of propositional variables, such that $(F_1\rightarrow F_2)\rightarrow F_3$ is a contradicti...

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381 views

GO Classes asked Mar 23, 2022

3 votes

2 answers

319

GO Classes 2023 | Weekly Quiz 3 | Question: 3
Here are some very useful ways of characterizing propositional formulas. Start by constructing a truth table for the formula and look at the column of values obtained. We say that the formula is: satisfiable if there is at least one $T$ ... Necessarily false: $F2$ is tautology $F3$ is tautology $F1$ is a contingency. $F1\wedge F3$ is contradiction.

Here are some very useful ways of characterizing propositional formulas. Start by constructing a truth table for the formula and look at the column of values obtained. We...

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764 views

GO Classes asked Mar 23, 2022

5 votes

2 answers

320

GO Classes 2023 | Weekly Quiz 3 | Question: 4
Let $A,\;B$ represent two propositions. Which of the following logical formulae is/are tautology? $A\wedge B\rightarrow (A\rightarrow B)$ $A\wedge \neg B \rightarrow \neg (A\rightarrow B)$ $\neg A\wedge B \rightarrow (A\rightarrow B)$ $\neg A\wedge \neg B \rightarrow (A\rightarrow B)$

Let $A,\;B$ represent two propositions.Which of the following logical formulae is/are tautology?$A\wedge B\rightarrow (A\rightarrow B)$$A\wedge \neg B \rightarrow \neg (A...

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520 views

GO Classes asked Mar 23, 2022

4 votes

1 answer

321

GO Classes 2023 | Weekly Quiz 3 | Question: 5
Let’s consider the interpretation $v$ where $v(p) = F, v(q) = T, v(r) = T.$ Which of the following propositional formulas are satisfied by $v$? $(p \rightarrow \neg q) \vee \neg(r \wedge q)$ $(\neg p \vee \neg q) \rightarrow (p \vee \neg r)$ $\neg(\neg p \rightarrow \neg q) \wedge r$ $\neg (\neg p \rightarrow q \wedge \neg r)$

Let’s consider the interpretation $v$ where $v(p) = F, v(q) = T, v(r) = T.$ Which of the following propositional formulas are satisfied by $v$?$(p \rightarrow \neg q) \...

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819 views

GO Classes asked Mar 23, 2022

5 votes

1 answer

322

GO Classes 2023 | Weekly Quiz 3 | Question: 6
If $F1$, and $F2$ are propositional formulae/expressions, over same set of propositional variables, such that $F1,F2$ both are contingencies, then which of the following is/are necessarily false(i.e. Never Possible): $F1 \vee F2$ is a ... a tautology. $F1 \vee F2$ is a contradiction $(F1 \rightarrow F2) \vee (F2 \rightarrow F1)$ is contingency.

If $F1$, and $F2$ are propositional formulae/expressions, over same set of propositional variables, such that $F1,F2$ both are contingencies, then which of the following ...

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751 views

GO Classes asked Mar 23, 2022

4 votes

1 answer

323

GO Classes 2023 | Weekly Quiz 3 | Question: 7
Let $p,q$ be two atomic propositional assertions. Then which of the following is/are false? $(p \rightarrow q) \vee (p \rightarrow \neg q)$ is a tautology. $(p \rightarrow q) \vee (q \rightarrow p)$ is a tautology. $(p \rightarrow q) \vee (q \rightarrow \neg p)$ is a tautology. $(p \rightarrow q) \vee (\neg q \rightarrow \neg p)$ is a tautology.

Let $p,q$ be two atomic propositional assertions. Then which of the following is/are false?$(p \rightarrow q) \vee (p \rightarrow \neg q)$ is a tautology.$(p \rightarrow ...

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417 views

GO Classes asked Mar 23, 2022

3 votes

2 answers

324

GO Classes 2023 | Weekly Quiz 3 | Question: 9
Consider the statement form $S$ : $\left((p\Rightarrow (q\Rightarrow r)) \Leftrightarrow ((p\wedge q)\Rightarrow r)\right)\wedge \sim p\wedge \sim q\wedge \sim r$ Which of the following is true ? $S$ is a tautology $S$ is a Contradiction $S$ is a Contingency $S$ is equivalent to $ \sim (p\vee q\vee r)$.

Consider the statement form $S$ :$\left((p\Rightarrow (q\Rightarrow r)) \Leftrightarrow ((p\wedge q)\Rightarrow r)\right)\wedge \sim p\wedge \sim q\wedge \sim r$Which of ...

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664 views

GO Classes asked Mar 23, 2022

4 votes

3 answers

325

GO Classes 2023 | Weekly Quiz 3 | Question: 10
The shortest formula in propositional logic is $’p’$ where $p$ is a propositional atom. Which one of the following statements is TRUE? The formula $p$ is valid and satisfiable. The formula $p$ is invalid and unsatisfiable. The formula $p$ is invalid and satisfiable. The formula $p$ is valid and unsatisfiable.

The shortest formula in propositional logic is $’p’$ where $p$ is a propositional atom.Which one of the following statements is TRUE?The formula $p$ is valid and sati...

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699 views

GO Classes asked Mar 23, 2022

5 votes

2 answers

326

GO Classes 2023 | Weekly Quiz 3 | Question: 12
An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie. You encounter two people $A$ and $B$. What are $A$ and $B$ if: $A$ says $B$ is a knight and $B$ says The two of us are opposite types ? $A$ is ... , $B$ is knight $B$ is knight, $B$ is knave. $A$ is knave, $B$ is knave. $A$ is knave, $B$ is knight

An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie. You encounter two people $A$ and $B$. What are $A$ and $B$ if: $A$...

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588 views

GO Classes asked Mar 23, 2022

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