Mathematical Reasoning · Mathematics · JEE Main

The statement $$(p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q))$$ is equivalent to _________.

The negation of the statement $$((A \wedge(B \vee C)) \Rightarrow(A \vee B)) \Rightarrow A$$ is

Among the two statements

$$(\mathrm{S} 1):(p \Rightarrow q) \wedge(p \wedge(\sim q))$$ is a contradiction and

$$(\mathrm{S} 2):(p \wedge q) \vee((\sim p) \wedge q) \vee(p \wedge(\sim q)) \vee((\sim p) \wedge(\sim q))$$ is a tautology

The converse of $$((\sim p) \wedge q) \Rightarrow r$$ is

The statement $$\sim[p \vee(\sim(p \wedge q))]$$ is equivalent to :

The negation of the statement $$(p \vee q) \wedge (q \vee ( \sim r))$$ is :

The negation of $$(p \wedge(\sim q)) \vee(\sim p)$$ is equivalent to :

Negation of $$(p \Rightarrow q) \Rightarrow(q \Rightarrow p)$$ is :

Among the statements

(S1) : $$(p \Rightarrow q) \vee((\sim p) \wedge q)$$ is a tautology

(S2) : $$(q \Rightarrow p) \Rightarrow((\sim p) \wedge q)$$ is a contradiction

Statement $$\mathrm{(P \Rightarrow Q) \wedge(R \Rightarrow Q)}$$ is logically equivalent to :

Which of the following statements is a tautology?

The negation of the expression $$q \vee \left( {( \sim \,q) \wedge p} \right)$$ is equivalent to

$$(\mathrm{S} 1)~(p \Rightarrow q) \vee(p \wedge(\sim q))$$ is a tautology

$$(\mathrm{S} 2)~((\sim p) \Rightarrow(\sim q)) \wedge((\sim p) \vee q)$$ is a contradiction.

Then

Consider the following statements:

P : I have fever

Q: I will not take medicine

$\mathrm{R}$ : I will take rest.

The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to :

Among the statements :

$$(\mathrm{S} 1)~((\mathrm{p} \vee \mathrm{q}) \Rightarrow \mathrm{r}) \Leftrightarrow(\mathrm{p} \Rightarrow \mathrm{r})$$

$$(\mathrm{S} 2)~((\mathrm{p} \vee \mathrm{q}) \Rightarrow \mathrm{r}) \Leftrightarrow((\mathrm{p} \Rightarrow \mathrm{r}) \vee(\mathrm{q} \Rightarrow \mathrm{r}))$$

If $$p,q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p,q$$ and $$r$$ makes the logical expression $$\left\{ {(p \vee q) \wedge \left( {( \sim p) \vee r} \right)} \right\} \to \left( {( \sim q) \vee r} \right)$$ false?

Let $$\Delta ,\nabla \in \{ \wedge , \vee \} $$ be such that $$\mathrm{(p \to q)\Delta (p\nabla q)}$$ is a tautology. Then

The statement $$\left( {p \wedge \left( { \sim q} \right)} \right) \Rightarrow \left( {p \Rightarrow \left( { \sim q} \right)} \right)$$ is

Let p and q be two statements. Then $$ \sim \left( {p \wedge (p \Rightarrow \, \sim q)} \right)$$ is equivalent to

The compound statement $$\left( { \sim (P \wedge Q)} \right) \vee \left( {( \sim P) \wedge Q} \right) \Rightarrow \left( {( \sim P) \wedge ( \sim Q)} \right)$$ is equivalent to

The statement $$(p \Rightarrow q) \vee(p \Rightarrow r)$$ is NOT equivalent to

The statement $$(p \wedge q) \Rightarrow(p \wedge r)$$ is equivalent to :

Let

$$\mathrm{p}$$ : Ramesh listens to music.

$$\mathrm{q}$$ : Ramesh is out of his village.

$$\mathrm{r}$$ : It is Sunday.

$$\mathrm{s}$$ : It is Saturday.

Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as

Let the operations $$*, \odot \in\{\wedge, \vee\}$$. If $$(\mathrm{p} * \mathrm{q}) \odot(\mathrm{p}\, \odot \sim \mathrm{q})$$ is a tautology, then the ordered pair $$(*, \odot)$$ is :

If the truth value of the statement $$(P \wedge(\sim R)) \rightarrow((\sim R) \wedge Q)$$ is F, then the truth value of which of the following is $$\mathrm{F}$$ ?

$$(p \wedge r) \Leftrightarrow(p \wedge(\sim q))$$ is equivalent to $$(\sim p)$$ when $$r$$ is

Negation of the Boolean expression $$p \Leftrightarrow(q \Rightarrow p)$$ is

The statement $$(\sim(\mathrm{p} \Leftrightarrow \,\sim \mathrm{q})) \wedge \mathrm{q}$$ is :

Consider the following statements:

P : Ramu is intelligent.

Q : Ramu is rich.

R : Ramu is not honest.

The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:

Which of the following statements is a tautology ?

The conditional statement

$$((p \wedge q) \to (( \sim p) \vee r)) \vee ((( \sim p) \vee r) \to (p \wedge q))$$ is :

Negation of the Boolean statement (p $$\vee$$ q) $$\Rightarrow$$ (($$\sim$$ r) $$\vee$$ p) is equivalent to :

Let $$\Delta$$ $$\in$$ {$$\wedge$$, $$\vee$$, $$\Rightarrow$$, $$\Leftrightarrow$$} be such that (p $$\wedge$$ q) $$\Delta$$ ((p $$\vee$$ q) $$\Rightarrow$$ q) is a tautology. Then $$\Delta$$ is equal to :

Let p, q, r be three logical statements. Consider the compound statements

$${S_1}:(( \sim p) \vee q) \vee (( \sim p) \vee r)$$ and

$${S_2}:p \to (q \vee r)$$

Then, which of the following is NOT true?

Which of the following statement is a tautology?

The boolean expression $$( \sim (p \wedge q)) \vee q$$ is equivalent to :

Let r $$\in$$ {p, q, $$\sim$$p, $$\sim$$q} be such that the logical statement

r $$\vee$$ ($$\sim$$p) $$\Rightarrow$$ (p $$\wedge$$ q) $$\vee$$ r

is a tautology. Then r is equal to :

Let $$\Delta$$, $$\nabla $$ $$\in$$ {$$\wedge$$, $$\vee$$} be such that p $$\nabla$$ q $$\Rightarrow$$ ((p $$\Delta$$ q) $$\nabla$$ r) is a tautology. Then (p $$\nabla$$ q) $$\Delta$$ r is logically equivalent to :

The negation of the Boolean expression (($$\sim$$ q) $$\wedge$$ p) $$\Rightarrow$$ (($$\sim$$ p) $$\vee$$ q) is logically equivalent to :

Consider the following two propositions:

$$P1: \sim (p \to \sim q)$$

$$P2:(p \wedge \sim q) \wedge (( \sim p) \vee q)$$

If the proposition $$p \to (( \sim p) \vee q)$$ is evaluated as FALSE, then :

Consider the following statements:

A : Rishi is a judge.

B : Rishi is honest.

C : Rishi is not arrogant.

The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

The number of choices for $$\Delta \in \{ \wedge , \vee , \Rightarrow , \Leftrightarrow \} $$, such that

$$(p\Delta q) \Rightarrow ((p\Delta \sim q) \vee (( \sim p)\Delta q))$$ is a tautology, is :

The number of ordered triplets of the truth values of $$p, q$$ and $$r$$ such that the truth value of the statement $$(p \vee q) \wedge(p \vee r) \Rightarrow(q \vee r)$$ is True, is equal to ___________.

The maximum number of compound propositions, out of p$$\vee$$r$$\vee$$s, p$$\vee$$r$$\vee$$$$\sim$$s, p$$\vee$$$$\sim$$q$$\vee$$s, $$\sim$$p$$\vee$$$$\sim$$r$$\vee$$s, $$\sim$$p$$\vee$$$$\sim$$r$$\vee$$$$\sim$$s, $$\sim$$p$$\vee$$q$$\vee$$$$\sim$$s, q$$\vee$$r$$\vee$$$$\sim$$s, q$$\vee$$$$\sim$$r$$\vee$$$$\sim$$s, $$\sim$$p$$\vee$$$$\sim$$q$$\vee$$$$\sim$$s that can be made simultaneously true by an assignment of the truth values to p, q, r and s, is equal to __________.

The statement $$B \Rightarrow \left( {\left( { \sim A} \right) \vee B} \right)$$ is equivalent to :