Recent questions tagged mathematical-logic

3 votes

2 answers

151

Discrete Mathematics | Propositional Logic | Test 1 | Question: 15
A compound sentence is a $\textit{tautology}$ if it is true independently of the truth values of its component atomic sentences. A sentence is $\textit{atomic}$ if it contains no sentential connectives. A sentence $P$ ... ) $\neg Q \rightarrow \neg P$ $Q \rightarrow P$ $P \rightarrow Q$ $\neg P \wedge Q$

A compound sentence is a $\textit{tautology}$ if it is true independently of the truth values of its component atomic sentences. A sentence is $\textit{atomic}$ if it con...

ankitgupta.1729

416 views

ankitgupta.1729 asked Apr 11, 2023

4 votes

2 answers

152

GO Classes 2025 | Weekly Quiz 7 | Propositional Logic | Question: 1
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. $(p \rightarrow q) \vee (q \rightarrow p)$ ... $(p \rightarrow q) \vee (\neg q \rightarrow \neg p)$ 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 ...

512 views

GO Classes asked Apr 5, 2023

9 votes

3 answers

153

GO Classes 2025 | Weekly Quiz 7 | Propositional Logic | Question: 2
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) \...

541 views

GO Classes asked Apr 5, 2023

10 votes

6 answers

154

GO Classes 2025 | Weekly Quiz 7 | 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...

602 views

GO Classes asked Apr 5, 2023

7 votes

2 answers

155

GO Classes 2025 | Weekly Quiz 7 | Propositional Logic | Question: 4
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. $\text{Argument 2:}$ ... ? 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,...

414 views

GO Classes asked Apr 5, 2023

7 votes

2 answers

156

GO Classes 2025 | Weekly Quiz 7 | 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, then $G$ ... $(F \rightarrow G) \wedge (F \rightarrow \neg G)$ 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...

680 views

GO Classes asked Apr 5, 2023

4 votes

2 answers

157

GO Classes 2025 | Weekly Quiz 7 | 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 $P$ and $Q,$ ... $R,$ 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 ...

310 views

GO Classes asked Apr 5, 2023

11 votes

2 answers

158

GO Classes 2024 | Weekly Quiz 7 | Propositional Logic | Question: 13
Consider the following list of $137$ statements: $\left(S_1\right): $ There is exactly $1$ false statement in this list. $\left(S_2\right): $ There are exactly $2$ ... true. It is possible that more than one statement in this list is true. Exactly one statement in this list is true.

Consider the following list of $137$ statements:$\left(S_1\right): $ There is exactly $1$ false statement in this list.$\left(S_2\right): $ There are exactly $2$ false st...

586 views

GO Classes asked Apr 5, 2023

7 votes

2 answers

159

GO Classes 2025 | Weekly Quiz 7 | Propositional Logic | Question: 7
Which of the following logical arguments is/are valid? $\begin{aligned} & P \rightarrow(Q \rightarrow R) \\ & \neg R \\ \therefore & \neg P \\ & \end{aligned}$ ...

Which of the following logical arguments is/are valid?$\begin{aligned} & P \rightarrow(Q \rightarrow R) \\ & \neg R \\ \therefore & \neg P \\ & \end{aligned}$$\begin{alig...

433 views

GO Classes asked Apr 5, 2023

12 votes

2 answers

160

GO Classes 2025 | Weekly Quiz 7 | 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 ...

413 views

GO Classes asked Apr 5, 2023

20 votes

6 answers

161

GO Classes 2025 | Weekly Quiz 7 | 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 $p_1, p_2, \ldots, p_n, r$, and conclusion $q$ ... 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...

806 views

GO Classes asked Apr 5, 2023

0 votes

1 answer

162

Discrete Math and its applications by Kenneth Rosen 7th edition Section 6:Counting
Can this question be explained in detail as i have trouble understanding it? "How many 4-permutations of the positive integers not exceeding 100 contain three consecutive integers k, k + 1, k + 2, in the ... the solution manual says we have to account for double counting but I don't understand how? Why 97?

Can this question be explained in detail as i have trouble understanding it? "How many 4-permutations of the positive integers not exceeding 100 contain three consecutive...

discretemather1

489 views

discretemather1 asked Apr 2, 2023

0 votes

1 answer

163

Discrete-mathematics | Mathematical-logic
Which of the following predicate is/are valid? S1: $\forall x(A(x)\Leftrightarrow B(x))\Rightarrow \exists x(A(x)\Rightarrow B(x))$ S2: $\exists x(A(x)\vee B(x))\Leftrightarrow (\exists xA(x)\vee\exists xB(x))$ A. only S1 B. only S2 C. Both D. None

Which of the following predicate is/are valid? S1: $\forall x(A(x)\Leftrightarrow B(x))\Rightarrow \exists x(A(x)\Rightarrow B(x))$ S2: $\exists x(A(x)\vee B(x))\Le...

370 views

Jay Patel 009 asked Mar 31, 2023

0 votes

1 answer

164

Dearangements in permutations and combinations
How many derangements of {1, 2, 3, 4, 5, 6, 7} begin with the integers 1, 2, & 3 in some order

How many derangements of {1, 2, 3, 4, 5, 6, 7} begin with the integers 1, 2, & 3 in some order

433 views

Aj1202 asked Mar 30, 2023

5 votes

2 answers

165

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 1
What is logically equivalent to "If Kareena and Parineeti go to the shopping mall then it is raining": If Kareena and Parineeti do not go to the shopping mall then it is not raining. If Kareena and Parineeti do not ... to the shopping mall. If it is not raining then Kareena and Parineeti do not go to the shopping mall.

What is logically equivalent to "If Kareena and Parineeti go to the shopping mall then it is raining":If Kareena and Parineeti do not go to the shopping mall then it is n...

397 views

GO Classes asked Mar 26, 2023

7 votes

3 answers

166

GO Classes 2025 | Weekly Quiz 6 | 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...

333 views

GO Classes asked Mar 26, 2023

4 votes

3 answers

167

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 3
Select the proposition that is a tautology. $(p \wedge q) \rightarrow \neg p$ $(p \vee q) \rightarrow p$ $(p \wedge q) \leftrightarrow p$ $(p \wedge q) \rightarrow p$

Select the proposition that is a tautology.$(p \wedge q) \rightarrow \neg p$$(p \vee q) \rightarrow p$$(p \wedge q) \leftrightarrow p$$(p \wedge q) \rightarrow p$

348 views

GO Classes asked Mar 26, 2023

6 votes

1 answer

168

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 4
Select the proposition that is a contradiction. $\neg(p \vee q) \wedge p$ $(p \vee q) \wedge p$ $(\neg p \wedge q) \leftrightarrow p$ $(\neg p \wedge q) \rightarrow p$

Select the proposition that is a contradiction.$\neg(p \vee q) \wedge p$$(p \vee q) \wedge p$$(\neg p \wedge q) \leftrightarrow p$$(\neg p \wedge q) \rightarrow p$

289 views

GO Classes asked Mar 26, 2023

7 votes

3 answers

169

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 5
In propositional logic, given $P$ and $P \rightarrow Q$, we can infer ________ $\sim Q$ $Q$ $P \wedge Q$ $\sim P \wedge Q$

In propositional logic, given $P$ and $P \rightarrow Q$, we can infer ________$\sim Q$$Q$$P \wedge Q$$\sim P \wedge Q$

562 views

GO Classes asked Mar 26, 2023

11 votes

3 answers

170

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 6
Which of the following statements is true? The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2} \wedge \dots \wedge S_{n}\rightarrow S$ is satisfiable. The sentence $S$ is a logical ... $S_{1}\wedge S_{2}\wedge \dots \wedge S_{n}\wedge S$ is inconsistent.

Which of the following statements is true?The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2} \wedge \dots \wedge S_{n}\rig...

855 views

GO Classes asked Mar 26, 2023

14 votes

2 answers

171

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 7
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 $F_1$ ... $F_3$ 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...

367 views

GO Classes asked Mar 26, 2023

35 votes

3 answers

172

GO Classes 2025 | Weekly Quiz 6 | 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 $\psi$ is ... 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 $\varphi$ $\Rightarrow$...

731 views

GO Classes asked Mar 26, 2023

21 votes

1 answer

173

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 9
Recall the definitions of valid, satisfiable, and unsatisfiable in the propositional logic. What is the relationship between the above concepts? If a formula $\phi$ is valid then $\phi$ is satisfiable; and if $\phi$ ... satisfiable, then $\phi$ is invalid; and if $\neg \phi$ is satisfiable then $\phi$ is valid.

Recall the definitions of valid, satisfiable, and unsatisfiable in the propositional logic.What is the relationship between the above concepts?If a formula $\phi$ is vali...

520 views

GO Classes asked Mar 26, 2023

16 votes

4 answers

174

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 10
In propositional logic if $\left ( P \rightarrow Q \right )\wedge \left ( R \rightarrow S \right )$ and $\left ( P \vee R \right )$ are two premises such that $\begin{array}{c} (P \to Q) \wedge (R \to S) \\ P \vee R \\ \hline Y \\ \hline \end{array}$ $Y$ is the premise : $P \vee R$ $P \vee S$ $Q \vee R$ $Q \vee S$

In propositional logic if $\left ( P \rightarrow Q \right )\wedge \left ( R \rightarrow S \right )$ and $\left ( P \vee R \right )$ are two premises such that$$\begin{arr...

849 views

GO Classes asked Mar 26, 2023

26 votes

1 answer

175

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 11
A set of propositions is called a system specification. System specification is consistent if they do not contain conflicting requirements that could be used to derive a contradiction. When specifications are not consistent, there ... . Which of the above system specifications are consistent? Only $1$ Only $2$ Both None

A set of propositions is called a system specification.System specification is consistent if they do not contain conflicting requirements that could be used to derive a c...

1.0k views

GO Classes asked Mar 26, 2023

34 votes

3 answers

176

GO Classes 2025 | Weekly Quiz 6 | 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...

1.2k views

GO Classes asked Mar 26, 2023

5 votes

2 answers

177

GO Classes 2025 | Weekly Quiz 6 | 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...

393 views

GO Classes asked Mar 26, 2023

8 votes

3 answers

178

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 14
Consider a proposition given as: $x \geq 6$, if $x^2 \geq 25 $ and and its proof as: If $x \geq 6$, then $x^2 =x.x \geq 6.6 = 36 \geq 25$ Which of the following is correct with respect to the given proposition and its ... what is to be shown The proof is correct and there is nothing wrong $ a$ only $c$ only $a$ and $b$ $b$ only

Consider a proposition given as:$x \geq 6$, if $x^2 \geq 25 $ and and its proof as:If $x \geq 6$, then $x^2 =x.x \geq 6.6 = 36 \geq 25$Which of the following is correct w...

596 views

GO Classes asked Mar 26, 2023

10 votes

1 answer

179

GO Classes 2025 | Weekly Quiz 6 | Propositional Logic | Question: 15
If the bank receipt is forged, then Mr. M is liable. If Mr. M is liable, he will go bankrupt. If the bank will loan him money, he will not go bankrupt. The bank will loan him money. Which of the following can ... from the above statements? Mr. M is liable The receipt is not forged Mr. M will go bankrupt The bank will go bankrupt

If the bank receipt is forged, then Mr. M is liable.If Mr. M is liable, he will go bankrupt.If the bank will loan him money, he will not go bankrupt.The bank will loan hi...

302 views

GO Classes asked Mar 26, 2023

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