Question 1: Palette 1/120

JEE Mains 2023 MCQ

Negation of $p \wedge(q \wedge \sim(p \wedge q))$ is :

A

$(\sim(p \wedge q)) \wedge q$

B

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

C

$p \vee q$

D

$\sim(p \vee q)$

Question 2: Palette 2/120

JEE Mains 2023 MCQ

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

A

$(\sim p) \vee(\sim q)$

B

$p \vee(\sim q)$

C

$\mathrm{p} \vee \mathrm{q}$

D

$(\sim p) \vee q$

Question 3: Palette 3/120

JEE Mains 2023 MCQ

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

A

equivalent to $B ~\vee \sim C$

B

equivalent to $\sim A$

C

equivalent to $\sim C$

D

a fallacy

Question 4: Palette 4/120

JEE Mains 2023 MCQ

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

A

both are false.

B

only (S1) is true.

C

both are true.

D

only (S2) is true.

Question 5: Palette 5/120

JEE Mains 2023 MCQ

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

A

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

B

$(\sim \mathrm{r}) \Rightarrow \mathrm{p} \wedge \mathrm{q}$

C

$(\mathrm{p} \vee(\sim \mathrm{q})) \Rightarrow(\sim \mathrm{r})$

D

$(\sim \mathrm{r}) \Rightarrow((\sim \mathrm{p}) \wedge \mathrm{q})$

Question 6: Palette 6/120

JEE Mains 2023 MCQ

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

A

$(\sim(p \wedge q)) \wedge q$

B

$\sim(p \vee q)$

C

$(p \wedge q) \wedge(\sim p)$

D

$\sim(p \wedge q)$

Question 7: Palette 7/120

JEE Mains 2023 MCQ

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

A

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

B

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

C

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

D

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

Question 8: Palette 8/120

JEE Mains 2023 MCQ

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

A

$p \wedge q$

B

$p \wedge(\sim q)$

C

$p \wedge(q \wedge(\sim p))$

D

$p \vee(q \vee(\sim p))$

Question 9: Palette 9/120

JEE Mains 2023 MCQ

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

A

$(\sim q) \wedge p$

B

$q \wedge(\sim p)$

C

$p \vee(\sim q)$

D

$(\sim p) \vee q$

Question 10: Palette 10/120

JEE Mains 2023 MCQ

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

A

neither (S1) and (S2) is True

B

only (S2) is True

C

both $(\mathrm{S} 1)$ and $(\mathrm{S} 2)$ are True

D

only (S1) is True

Question 11: Palette 11/120

JEE Mains 2023 MCQ

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

A

$(P \Rightarrow R) \wedge(Q \Rightarrow R)$

B

$(P \Rightarrow R) \vee(Q \Rightarrow R)$

C

$(P \wedge R) \Rightarrow Q$

D

$(P \vee R) \Rightarrow Q$

Question 12: Palette 12/120

JEE Mains 2023 MCQ

Which of the following statements is a tautology?

A

$\mathrm{p\vee(p\wedge q)}$

B

$(\mathrm{p\wedge(p\to q))\to\,\sim q}$

C

$\mathrm{p\to (p\wedge (p\to q))}$

D

$(\mathrm{p\wedge q)\to(\sim (p)\to q)}$

Question 13: Palette 13/120

JEE Mains 2023 MCQ

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

A

$( \sim \,p) \wedge ( \sim \,q)$

B

$( \sim \,p) \vee q$

C

$p \wedge ( \sim \,q)$

D

$( \sim \,p) \vee ( \sim \,q)$

Question 14: Palette 14/120

JEE Mains 2023 MCQ

The number of values of $\mathrm{r} \in\{\mathrm{p}, \mathrm{q}, \sim \mathrm{p}, \sim \mathrm{q}\}$ for which $((\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \vee \mathrm{q})) \wedge((\mathrm{p} \wedge \mathrm{r}) \Rightarrow \mathrm{q})$ is a tautology, is :

A

2

B

1

C

4

D

3

Question 15: Palette 15/120

JEE Mains 2023 MCQ

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

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

Then

A

only (S2) is correct

B

both (S1) and (S2) are correct

C

only (S1) is correct

D

both (S1) and (S2) are wrong

Question 16: Palette 16/120

JEE Mains 2023 MCQ

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 :

A

$((\sim P) \vee \sim Q) \wedge((\sim P) \vee \sim R)$

B

$(P \vee \sim Q) \wedge(P \vee \sim R)$

C

$((\sim P) \vee \sim Q) \wedge((\sim P) \vee R)$

D

$(P \vee Q) \wedge((\sim P) \vee R)$

Question 17: Palette 17/120

JEE Mains 2023 MCQ

Among the statements :

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

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

A

only (S1) is a tautology

B

neither (S1) nor (S2) is a tautology

C

both (S1) and (S2) are tautologies

D

only (S2) is a tautology

Question 18: Palette 18/120

JEE Mains 2023 MCQ

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?

A

$p = F,q = T,r = F$

B

$p = T,q = T,r = F$

C

$p = T,q = F,r = T$

D

$p = T,q = F,r = F$

Question 19: Palette 19/120

JEE Mains 2023 MCQ

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

A

$\Delta = \vee ,\nabla = \vee $

B

$\Delta = \vee ,\nabla = \wedge $

C

$\Delta = \wedge ,\nabla = \wedge $

D

$\Delta = \wedge ,\nabla = \vee $

Question 20: Palette 20/120

JEE Mains 2023 MCQ

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

A

a tautology

B

equivalent to $\left( { \sim p} \right) \vee \left( { \sim q} \right)$

C

a contradiction

D

$p \vee q$

Question 21: Palette 21/120

JEE Mains 2023 MCQ

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

A

$\left( { \sim p} \right) \vee q$

B

$p \vee \left( {p \wedge ( \sim q)} \right)$

C

$p \vee \left( {p \wedge q} \right)$

D

$p \vee \left( {\left( { \sim p} \right) \wedge q} \right)$

Question 22: Palette 22/120

JEE Mains 2023 MCQ

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

A

$(( \sim P) \vee Q) \wedge ( \sim Q)$

B

$( \sim Q) \vee P$

C

$(( \sim P) \vee Q) \wedge (( \sim Q) \vee P)$

D

$( \sim P) \vee Q$

Question 23: Palette 23/120

JEE Mains 2023 MSQ

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

A

$B \Rightarrow \left( {\left( { \sim A} \right) \Rightarrow B} \right)$

B

$A \Rightarrow \left( {A \Leftrightarrow B} \right)$

C

$A \Rightarrow \left( {\left( { \sim A} \right) \Rightarrow B} \right)$

D

$B \Rightarrow \left( {A \Rightarrow B} \right)$

Question 24: Palette 24/120

JEE Mains 2022 MCQ

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

A

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

B

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

C

$p \Rightarrow(q \vee r)$

D

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

Question 25: Palette 25/120

JEE Mains 2022 MCQ

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

A

$q \Rightarrow(p \wedge r)$

B

$p\Rightarrow(\mathrm{p} \wedge \mathrm{r})$

C

$(\mathrm{p} \wedge \mathrm{r}) \Rightarrow(\mathrm{p} \wedge \mathrm{q})$

D

$(p \wedge q) \Rightarrow r$

Question 26: Palette 26/120

JEE Mains 2022 MCQ

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

A

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

B

$(\mathrm{q} \wedge(\mathrm{r} \vee \mathrm{s})) \Rightarrow \mathrm{p}$

C

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

D

$\mathrm{p} \Rightarrow((\sim \mathrm{q}) \wedge(\mathrm{r} \vee \mathrm{s}))$

Question 27: Palette 27/120

JEE Mains 2022 MCQ

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 :

A

$(\vee, \wedge)$

B

$(\vee, \vee)$

C

$(\wedge, \wedge)$

D

$(\wedge, \vee)$

Question 28: Palette 28/120

JEE Mains 2022 MCQ

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 $\mathrm{F}$ ?

A

$\mathrm{P} \vee \mathrm{Q} \rightarrow \,\sim \mathrm{R}$

B

$\mathrm{R} \vee \mathrm{Q} \rightarrow \,\sim \mathrm{P}$

C

$\sim(\mathrm{P} \vee \mathrm{Q}) \rightarrow \sim \mathrm{R}$

D

$\sim(\mathrm{R} \vee \mathrm{Q}) \rightarrow \,\sim \mathrm{P}$

Question 29: Palette 29/120

JEE Mains 2022 MCQ

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

A

$p$

B

$\sim p$

C

$q$

D

$\sim q$

Question 30: Palette 30/120

JEE Mains 2022 MCQ

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

A

$(\sim p) \wedge q$

B

$p \wedge(\sim q)$

C

$(\sim p) \vee(\sim q)$

D

$(\sim p) \wedge(\sim q)$

Question 31: Palette 31/120

JEE Mains 2022 MCQ

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

A

a tautology

B

a contradiction

C

equivalent to $(p \Rightarrow q) \wedge q$

D

equivalent to $(p \Rightarrow q) \wedge p$

Question 32: Palette 32/120

JEE Mains 2022 MCQ

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:

A

$((P \wedge(\sim R)) \wedge Q) \wedge((\sim Q) \wedge((\sim P) \vee R))$

B

$((P \wedge R) \wedge Q) \vee((\sim Q) \wedge((\sim P) \vee(\sim R)))$

C

$((P \wedge R) \wedge Q) \wedge((\sim Q) \wedge((\sim P) \vee(\sim R)))$

D

$((P \wedge(\sim R)) \wedge Q) \vee((\sim Q) \wedge((\sim P) \vee R))$

Question 33: Palette 33/120

JEE Mains 2022 MCQ

Which of the following statements is a tautology ?

A

$((\sim \mathrm{p}) \vee \mathrm{q}) \Rightarrow \mathrm{p}$

B

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

C

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

D

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

Question 34: Palette 34/120

JEE Mains 2022 MCQ

The conditional statement

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

A

a tautology

B

a contadiction

C

equivalent to $p \wedge q$

D

equivalent to $( \sim p) \vee r$

Question 35: Palette 35/120

JEE Mains 2022 MCQ

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

A

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

B

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

C

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

D

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

Question 36: Palette 36/120

JEE Mains 2022 MCQ

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 :

A

$\wedge$

B

$\vee$

C

$\Rightarrow$

D

$\Leftrightarrow$

Question 37: Palette 37/120

JEE Mains 2022 MCQ

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?

A

If S₂ is True, then S₁ is True

B

If S₂ is False, then S₁ is False

C

If S₂ is False, then S₁ is True

D

If S₁ is False, then S₂ is False

Question 38: Palette 38/120

JEE Mains 2022 MCQ

Which of the following statement is a tautology?

A

$(( \sim q) \wedge p) \wedge q$

B

$(( \sim q) \wedge p) \wedge (p \wedge ( \sim p))$

C

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

D

$(p \wedge q) \wedge ( \sim p \wedge q))$

Question 39: Palette 39/120

JEE Mains 2022 MCQ

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

A

$q \to (p \wedge q)$

B

$p \to q$

C

$p \to (p \to q)$

D

$p \to (p \vee q)$

Question 40: Palette 40/120

JEE Mains 2022 MCQ

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 :

A

p

B

q

C

$\sim$p

D

$\sim$q

Question 41: Palette 41/120

JEE Mains 2022 MCQ

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 :

A

(p $\Delta$ r) $\vee$ q

B

(p $\Delta$ r) $\wedge$ q

C

(p $\wedge$ r) $\Delta$ q

D

(p $\nabla$ r) $\wedge$ q

Question 42: Palette 42/120

JEE Mains 2022 MCQ

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

A

$p \Rightarrow q$

B

$q \Rightarrow p$

C

$ \sim (p \Rightarrow q)$

D

$ \sim (q \Rightarrow p)$

Question 43: Palette 43/120

JEE Mains 2022 MCQ

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 :

A

P1 is TRUE and P2 is FALSE

B

P1 is FALSE and P2 is TRUE

C

Both P1 and P2 are FALSE

D

Both P1 and P2 are TRUE

Question 44: Palette 44/120

JEE Mains 2022 MCQ

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

A

B $\to$ (A $\vee$ C)

B

($\sim$B) $\wedge$ (A $\wedge$ C)

C

B $\to$ (($\sim$A) $\vee$ ($\sim$C))

D

B $\to$ (A $\wedge$ C)

Question 45: Palette 45/120

JEE Mains 2022 MCQ

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 :

A

1

B

2

C

3

D

4

Question 46: Palette 46/120

JEE Mains 2021 MCQ

Which of the following is equivalent to the Boolean expression p $\wedge$ $\sim$ q ?

A

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

B

$\sim$ p $\to$ $\sim$ q

C

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

D

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

Question 47: Palette 47/120

JEE Mains 2021 MCQ

Negation of the statement (p $\vee$ r) $\Rightarrow$ (q $\vee$ r) is :

A

p $\wedge$ $\sim$ q $\wedge$ $\sim$ r

B

$\sim$ p $\wedge$ q $\wedge$ $\sim$ 4

C

$\sim$ p $\wedge$ q $\wedge$ r

D

p $\wedge$ q $\wedge$ r

Question 48: Palette 48/120

JEE Mains 2021 MCQ

Let *, ▢ $\in${$\wedge$, $\vee$} be such that the Boolean expression (p * $\sim$ q) $\Rightarrow$ (p ▢ q) is a tautology. Then :

A

* = $\vee$, ▢ = $\vee$

B

* = $\wedge$, ▢ = $\wedge$

C

* = $\wedge$, ▢ = $\vee$

D

* = $\vee$, ▢ = $\wedge$

Question 49: Palette 49/120

JEE Mains 2021 MCQ

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

A

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

B

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

C

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

D

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

Question 50: Palette 50/120

JEE Mains 2021 MCQ

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

A

a tautology

B

equivalent to p $\to$ $\sim$ r

C

a fallacy

D

equivalent to q $\to$ $\sim$ r

Question 51: Palette 51/120

JEE Mains 2021 MCQ

Consider the two statements :

(S1) : (p $\to$ q) $ \vee $ ($ \sim $ q $\to$ p) is a tautology .

(S2) : (p $ \wedge $ $ \sim $ q) $ \wedge $ ($\sim$ p $\wedge$ q) is a fallacy.

Then :

A

only (S1) is true.

B

both (S1) and (S2) are false.

C

both (S1) and (S2) are true.

D

only (S2) is true.

Question 52: Palette 52/120

JEE Mains 2021 MCQ

If the truth value of the Boolean expression $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right)$ is false, then the truth values of the statements p, q, r respectively can be :

A

T F T

B

F F T

C

T F F

D

F T F

Question 53: Palette 53/120

JEE Mains 2021 MCQ

Which of the following is the negation of the statement "for all M > 0, there exists x$\in$S such that x $\ge$ M" ?

A

there exists M > 0, such that x < M for all x$\in$S

B

there exists M > 0, there exists x$\in$S such that x $\ge$ M

C

there exists M > 0, there exists x$\in$S such that x < M

D

there exists M > 0, such that x $\ge$ M for all x$\in$S

Question 54: Palette 54/120

JEE Mains 2021 MCQ

The compound statement $(P \vee Q) \wedge ( \sim P) \Rightarrow Q$ is equivalent to :

A

$P \vee Q$

B

$P \wedge \sim Q$

C

$ \sim (P \Rightarrow Q)$

D

$ \sim (P \Rightarrow Q) \Leftrightarrow P \wedge \sim Q$

Question 55: Palette 55/120

JEE Mains 2021 MCQ

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following :

A

The match will not be played and weather is not good and ground is wet.

B

If the match will not be played, then either weather is not good or ground is wet.

C

The match will be played and weather is not good or ground is wet.

D

The match will not be played or weather is good and ground is not wet.

Question 56: Palette 56/120

JEE Mains 2021 MCQ

The Boolean expression $(p \Rightarrow q) \wedge (q \Rightarrow \sim p)$ is equivalent to :

A

$ \sim $ q

B

q

C

p

D

$ \sim $ p

Question 57: Palette 57/120

JEE Mains 2021 MCQ

Which of the following Boolean expressions is not a tautology?

A

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

B

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

C

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

D

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

Question 58: Palette 58/120

JEE Mains 2021 MCQ

Consider the following three statements :

(A) If 3 + 3 = 7 then 4 + 3 = 8

(B) If 5 + 3 = 8 then earth is flat.

(C) If both (A) and (B) are true then 5 + 6 = 17.

Then, which of the following statements is correct?

A

(A) is false, but (B) and (C) re true

B

(A) and (C) are true while (B) is false

C

(A) is true while (B) and (C) are false

D

(A) and (B) are false while (C) is true

Question 59: Palette 59/120

JEE Mains 2021 MCQ

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

A

$q \Rightarrow p$

B

$p \Rightarrow q$

C

$ \sim q \Rightarrow p$

D

$p \Rightarrow \, \sim q$

Question 60: Palette 60/120

JEE Mains 2021 MCQ

If P and Q are two statements, then which of the following compound statement is a tautology?

A

((P $ \Rightarrow $ Q) $ \wedge $ $ \sim $ Q) $ \Rightarrow $ (P $ \wedge $ Q)

B

((P $ \Rightarrow $ Q) $ \wedge $ $ \sim $ Q) $ \Rightarrow $ Q

C

((P $ \Rightarrow $ Q) $ \wedge $ $ \sim $ Q) $ \Rightarrow $ P

D

((P $ \Rightarrow $ Q) $ \wedge $ $ \sim $ Q) $ \Rightarrow $ $ \sim $ P

Question 61: Palette 61/120

JEE Mains 2021 MCQ

If the Boolean expression $(p \wedge q) \odot (p \otimes q)$ is a tautology, then $ \odot $ and $ \otimes $ are respectively given by :

A

$ \vee , \to $

B

$ \to $, $ \to $

C

$ \wedge $, $ \vee $

D

$ \wedge $, $ \to $

Question 62: Palette 62/120

JEE Mains 2021 MCQ

If the Boolean expression (p $ \Rightarrow $ q) $ \Leftrightarrow $ (q * ($ \sim $p) is a tautology, then the boolean expression (p * ($ \sim $q)) is equivalent to :

A

q $ \Rightarrow $ p

B

p $ \Rightarrow $ q

C

p $ \Rightarrow $ $ \sim $ q

D

$ \sim $q $ \Rightarrow $ p

Question 63: Palette 63/120

JEE Mains 2021 MCQ

Which of the following Boolean expression is a tautology?

A

(p $ \wedge $ q) $ \vee $ (p $ \to $ q)

B

(p $ \wedge $ q) $ \vee $ (p $\vee$ q)

C

(p $ \wedge $ q) $ \to $ (p $ \to $ q)

D

(p $ \wedge $ q) $ \wedge $ (p $ \to $ q)

Question 64: Palette 64/120

JEE Mains 2021 MCQ

Let F₁(A, B, C) = (A $ \wedge $ $ \sim $ B) $ \vee $ [$\sim$C $\wedge$ (A $\vee$ B)] $\vee$ $\sim$ A and
F₂(A, B) = (A $\vee$ B) $\vee$ (B $ \to $ $\sim$A) be two logical expressions. Then :

A

Both F₁ and F₂ are not tautologies

B

F₁ and F₂ both are tautologies

C

F₁ is not a tautology but F₂ is a tautology

D

F₁ is a tautology but F₂ is not a tautology

Question 65: Palette 65/120

JEE Mains 2021 MCQ

The contrapositive of the statement "If you will work, you will earn money" is :

A

If you will not earn money, you will not work

B

If you will earn money, you will work

C

You will earn money, if you will not work

D

To earn money, you need to work

Question 66: Palette 66/120

JEE Mains 2021 MCQ

The statement A $ \to $ (B $ \to $ A) is equivalent to :

A

A $ \to $ (A $\mathrel{\mathop{\kern0pt\longleftrightarrow} \limits_{}} $ B)

B

A $ \to $ (A $ \vee $ B)

C

A $ \to $ (A $ \wedge $ B)

D

A $ \to $ (A $ \to $ B)

Question 67: Palette 67/120

JEE Mains 2021 MCQ

The negation of the statement

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

A

$p \vee \sim q$

B

$ \sim p \vee q$

C

$ \sim p \wedge q$

D

$p \wedge \sim q$

Question 68: Palette 68/120

JEE Mains 2021 MCQ

For the statements p and q, consider the following compound statements :

(a) $( \sim q \wedge (p \to q)) \to \sim p$

(b) $((p \vee q) \wedge \sim p) \to q$

Then which of the following statements is correct?

A

(b) is a tautology but not (a).

B

(a) and (b) both are not tautologies.

C

(a) and (b) both are tautologies.

D

(a) is a tautology but not (b).

Question 69: Palette 69/120

JEE Mains 2021 MCQ

The statement among the following that is a tautology is :

A

$B \to \left[ {A \wedge \left( {A \to B} \right)} \right]$

B

$\left[ {A \wedge \left( {A \to B} \right)} \right] \to B$

C

$\left[ {A \wedge \left( {A \vee B} \right)} \right]$

D

$\left[ {A \vee \left( {A \wedge B} \right)} \right]$

Question 70: Palette 70/120

JEE Mains 2020 MCQ

Consider the statement :
‘‘For an integer n, if n³ – 1 is even, then n is odd.’’
The contrapositive statement of this statement is :

A

For an integer n, if n is even, then n³ – 1 is even.

B

For an integer n, if n³ – 1 is not even, then n is not odd.

C

For an integer n, if n is odd, then n³ – 1 is even.

D

For an integer n, if n is even, then n³ – 1 is odd.

Question 71: Palette 71/120

JEE Mains 2020 MCQ

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

A

$p \wedge \sim q$

B

$ \sim $$p \vee \sim q$

C

$ \sim p \wedge q$

D

$ \sim p \wedge \sim q$

Question 72: Palette 72/120

JEE Mains 2020 MCQ

The statement
$\left( {p \to \left( {q \to p} \right)} \right) \to \left( {p \to \left( {p \vee q} \right)} \right)$ is :

A

a tautology

B

a contradiction

C

equivalent to (p $ \vee $ q) $ \wedge $ ($ \sim $ p)

D

equivalent to (p $ \wedge $ q) $ \vee $ ($ \sim $ q)

Question 73: Palette 73/120

JEE Mains 2020 MCQ

The negation of the Boolean expression x $ \leftrightarrow $ ~ y is equivalent to :

A

$\left( {x \wedge y} \right) \vee \left( { \sim x \wedge \sim y} \right)$

B

$\left( { \sim x \wedge y} \right) \vee \left( { \sim x \wedge \sim y} \right)$

C

$\left( {x \wedge y} \right) \wedge \left( { \sim x \vee \sim y} \right)$

D

$\left( {x \wedge \sim y} \right) \vee \left( { \sim x \wedge y} \right)$

Question 74: Palette 74/120

JEE Mains 2020 MCQ

Contrapositive of the statement :
‘If a function f is differentiable at a, then it is also continuous at a’, is:

A

If a function f is continuous at a, then it is not differentiable at a.

B

If a function f is not continuous at a, then it is differentiable at a.

C

If a function f is not continuous at a, then it is not differentiable at a.

D

If a function f is continuous at a, then it is differentiable at a.

Question 75: Palette 75/120

JEE Mains 2020 MCQ

Given the following two statements:

$\left( {{S_1}} \right):\left( {q \vee p} \right) \to \left( {p \leftrightarrow \sim q} \right)$ is a tautology

$\left( {{S_2}} \right): \,\,\sim q \wedge \left( { \sim p \leftrightarrow q} \right)$ is a fallacy. Then:

A

both (S₁) and (S₂) are not correct

B

only (S₁) is correct

C

only (S₂) is correct

D

both (S₁) and (S₂) are correct

Question 76: Palette 76/120

JEE Mains 2020 MCQ

Let p, q, r be three statements such that the truth value of
(p $ \wedge $ q) $ \to $ ($ \sim $q $ \vee $ r) is F. Then the truth values of p, q, r are respectively :

A

T, F, T

B

F, T, F

C

T, T, T

D

T, T, F

Question 77: Palette 77/120

JEE Mains 2020 MCQ

The proposition p $ \to $ ~ (p $ \wedge $ ~q) is equivalent to :

A

($ \sim $p) $ \vee $ q

B

q

C

($ \sim $p) $ \wedge $ q

D

($ \sim $p) $ \vee $ ($ \sim $q)

Question 78: Palette 78/120

JEE Mains 2020 MCQ

Which of the following is a tautology ?

A

$\left( { \sim p} \right) \wedge \left( {p \vee q} \right) \to q$

B

$\left( {q \to p} \right) \vee \sim \left( {p \to q} \right)$

C

$\left( {p \to q} \right) \wedge \left( {q \to p} \right)$

D

$\left( { \sim q} \right) \vee \left( {p \wedge q} \right) \to q$

Question 79: Palette 79/120

JEE Mains 2020 MCQ

The contrapositive of the statement
"If I reach the station in time, then I will catch the train" is :

A

If I will catch the train, then I reach the station in time.

B

If I do not reach the station in time, then I will not catch the train.

C

If I will not catch the train, then I do not reach the station in time.

D

If I do not reach the station in time, then I will catch the train.

Question 80: Palette 80/120

JEE Mains 2020 MCQ

If p $ \to $ (p $ \wedge $ ~q) is false, then the truth values of p and q are respectively :

A

T, T

B

T, F

C

F, T

D

F, F

Question 81: Palette 81/120

JEE Mains 2020 MCQ

Negation of the statement :

$\sqrt 5 $ is an integer or 5 is an irrational is :

A

$\sqrt 5 $ is not an integer and 5 is not irrational.

B

$\sqrt 5 $ is irrational or 5 is an integer.

C

$\sqrt 5 $ is an integer and 5 is irrational.

D

$\sqrt 5 $ is not an integer or 5 is not irrational.

Question 82: Palette 82/120

JEE Mains 2020 MCQ

Which of the following statements is a tautology?

A

~(p $ \wedge $ ~q) $ \to $ p $ \vee $ q

B

~(p $ \vee $ ~q) $ \to $ p $ \vee $ q

C

~(p $ \vee $ ~q) $ \to $ p $ \wedge $ q

D

p $ \vee $ (~q) $ \to $ p $ \wedge $ q

Question 83: Palette 83/120

JEE Mains 2020 MCQ

Which one of the following is a tautology?

A

P $ \wedge $ (P $ \vee $ Q)

B

P $ \vee $ (P $ \wedge $ Q)

C

Q $ \to $ (P $ \wedge $ (P $ \to $ Q))

D

(P $ \wedge $ (P $ \to $ Q)) $ \to $ Q

Question 84: Palette 84/120

JEE Mains 2020 MCQ

Let A, B, C and D be four non-empty sets. The contrapositive statement of "If A $ \subseteq $ B and B $ \subseteq $ D, then A $ \subseteq $ C" is :

A

If A ⊈ C, then A ⊈ B or B ⊈ D

B

If A ⊈ C, then A ⊈ B and B $ \subseteq $ D

C

If A $ \subseteq $ C, then B $ \subset $ A or D $ \subset $ B

D

If A ⊈ C, then A $ \subseteq $ B and B $ \subseteq $ D

Question 85: Palette 85/120

JEE Mains 2020 MCQ

The logical statement (p $ \Rightarrow $ q) $\Lambda $ ( q $ \Rightarrow $ ~p) is equivalent to :

A

q

B

$ \sim $p

C

p

D

$ \sim $q

Question 86: Palette 86/120

JEE Mains 2019 MCQ

The Boolean expression ~(p $ \Rightarrow $ (~q)) is equivalent to :

A

p $ \wedge $ q

B

q $ \Rightarrow $ ~p

C

p $ \vee $ q

D

(~p) $ \Rightarrow $ q

Question 87: Palette 87/120

JEE Mains 2019 MCQ

If the truth value of the statement p $ \to $ (~q $ \vee $ r) is false (F), then the truth values of the statements p, q, r are respectively :

A

T, F, T

B

F, T, T

C

T, T, F

D

T, F, F

Question 88: Palette 88/120

JEE Mains 2019 MCQ

The negation of the Boolean expression ~ s $ \vee $ (~r $ \wedge $ s) is equivalent to :

A

~ s $ \wedge $ ~ r

B

r

C

s $ \vee $ r

D

s $ \wedge $ r

Question 89: Palette 89/120

JEE Mains 2019 MCQ

Which one of the following Boolean expressions is a tautology?

A

(p $ \vee $ q) $ \wedge $ (~ p $ \vee $ ~ q)

B

(p $ \vee $ q) $ \vee $ ( p $ \vee $ ~ q)

C

(p $ \wedge $ q) $ \vee $ ( p $ \wedge $ ~ q)

D

(p $ \vee $ q) $ \wedge $ ( p $ \vee $ ~ q)

Question 90: Palette 90/120

JEE Mains 2019 MCQ

If p $ \Rightarrow $ (q $ \vee $ r) is false, then the truth values of p, q, r are respectively :-

A

F, F, F

B

T, F, F

C

F, T, T

D

T, T, F

Question 91: Palette 91/120

JEE Mains 2019 MCQ

For any two statements p and q, the negation of the expression
p $ \vee $ (~p $ \wedge $ q) is :

A

p$ \leftrightarrow $q

B

~p$ \wedge $~q

C

p$ \wedge $q

D

~p$ \vee $~q

Question 92: Palette 92/120

JEE Mains 2019 MCQ

Which one of the following statements is not a tautology?

A

(p $ \wedge $ q) $ \to $ (~ p) $ \vee $ q

B

(p $ \wedge $ q) $ \to $ p

C

( p $ \vee $ q) $ \to $ ( p $ \vee $ (~q))

D

p $ \to $ ( p $ \vee $ q)

Question 93: Palette 93/120

JEE Mains 2019 MCQ

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is :

A

If you are not a citizen of India, then you are not born in India.

B

If you are born in India, then you are not a citizen of India.

C

If you are a citizen of India, then you are born in India.

D

If you are not born in India, then you are not a citizen of India

Question 94: Palette 94/120

JEE Mains 2019 MCQ

The expression $ \sim $ ($ \sim $ p $ \to $ q) is logically equivalent to :

A

p $ \wedge $ q

B

$ \sim $ p $ \wedge $ $ \sim $ q

C

p $ \wedge $ $ \sim $ q

D

$ \sim $ p $ \wedge $ q

Question 95: Palette 95/120

JEE Mains 2019 MCQ

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

A

p $ \wedge $ q

B

p $ \wedge $ ($ \sim $ q)

C

p $ \vee $ ($ \sim $ q)

D

($ \sim $ p) $ \wedge $ ($ \sim $ q)

Question 96: Palette 96/120

JEE Mains 2019 MCQ

Contrapositive of the statement " If two numbers are not equal, then their squares are not equal." is :

A

If the squares of two numbers are equal, then the numbers are not equal

B

If the squares of two numbers are equal, then the numbers are equal

C

If the squares of two numbers are not equal, then the numbers are equal

D

If the squares of two numbers are not equal, then the numbers are not equal

Question 97: Palette 97/120

JEE Mains 2019 MCQ

If q is false and p $ \wedge $ q $ \leftrightarrow $ r is true, then which one of the following statements is a tautology ?

A

P $ \wedge $ r

B

(p $ \vee $ r) $ \to $ (p $ \wedge $ r)

C

p $ \vee $ r

D

(p $ \wedge $ r) $ \to $ (p $ \vee $ r)

Question 98: Palette 98/120

JEE Mains 2019 MCQ

Consider the following three statements :

P : 5 is a prime number

Q : 7 is a factor of 192

R : L.C.M. of 5 and 7 is 35

Then the truth value of which one of the following statements is true ?

A

(P $ \wedge $ Q) $ \vee $ ($ \sim $ R)

B

P $ \vee $ ($ \sim $ Q $ \wedge $ R)

C

(~ P) $ \wedge $ ($ \sim $ Q $ \wedge $ R)

D

($ \sim $ P) $ \vee $ (Q $ \wedge $ R)

Question 99: Palette 99/120

JEE Mains 2019 MCQ

Consider the statement : "P(n) : n² – n + 41 is prime". Then which one of the following is true ?

A

P(5) is false but P(3) is true

B

Both P(3) and P(5) are true

C

P(3) is false but P(5) is true

D

Both P(3) and P(5) are false

Question 100: Palette 100/120

JEE Mains 2019 MCQ

The logical statement

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

A

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

B

$ \sim $ p $ \vee $ r

C

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

D

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

Question 101: Palette 101/120

JEE Mains 2019 MCQ

If the Boolean expression
(p $ \oplus $ q) $\wedge$ (~ p $ \odot $ q) is equivalent
to p $\wedge$ q, where $ \oplus , \odot \in \left\{ { \wedge , \vee } \right\}$, then the
ordered pair $\left( { \oplus , \odot } \right)$ is :

A

$\left( { \vee , \wedge } \right)$

B

$\left( { \vee , \vee } \right)$

C

$\left( { \wedge , \vee } \right)$

D

$\left( { \wedge , \wedge } \right)$

Question 102: Palette 102/120

JEE Mains 2018 MCQ

If p $ \to $ ($ \sim $ p$ \vee $ $ \sim $ q) is false, then the truth values of p and q are respectively :

A

F, F

B

T, F

C

F, T

D

T, T

Question 103: Palette 103/120

JEE Mains 2018 MCQ

The Boolean expression

$ \sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$ is equvalent to :

A

${ \sim q}$

B

${ \sim p}$

C

p

D

q

Question 104: Palette 104/120

JEE Mains 2018 MCQ

Consider the following two statements :

Statement p :
The value of sin 120^o can be derived by taking $\theta = {240^o}$ in the equation
2sin${\theta \over 2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta } $

Statement q :
The angles A, B, C and D of any quadrilateral ABCD satisfy the equation
cos$\left( {{1 \over 2}\left( {A + C} \right)} \right) + \cos \left( {{1 \over 2}\left( {B + D} \right)} \right) = 0$

Then the truth values of p and q are respectively :

A

F, T

B

T, F

C

T, T

D

F, F

Question 105: Palette 105/120

JEE Mains 2018 MCQ

If (p $ \wedge $ $ \sim $ q) $ \wedge $ (p $ \wedge $ r) $ \to $ $ \sim $ p $ \vee $ q is false, then the truth values of $p, q$ and $r$ are, respectively :

A

F, T, F

B

T, F, T

C

T, T, T

D

F, F, F

Question 106: Palette 106/120

JEE Mains 2017 MCQ

Contrapositive of the statement

‘If two numbers are not equal, then their squares are not equal’, is :

A

If the squares of two numbers are equal, then the numbers are equal.

B

If the squares of two numbers are equal, then the numbers are not equal.

C

If the squares of two numbers are not equal, then the numbers are not equal.

D

If the squares of two numbers are not equal, then the numbers are equal.

Question 107: Palette 107/120

JEE Mains 2017 MCQ

The proposition $\left( { \sim p} \right) \vee \left( {p \wedge \sim q} \right)$ is equivalent to :

A

p $ \vee $ ~ q

B

p $ \to $ ~ q

C

p $ \wedge $ ~ q

D

q $ \to $ p

Question 108: Palette 108/120

JEE Mains 2017 MCQ

The following statement

$\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ is :

A

equivalent to ${ \sim p \to q}$

B

equivalent to ${p \to \sim q}$

C

a fallacy

D

a tautology

Question 109: Palette 109/120

JEE Mains 2016 MCQ

The contrapositive of the following statement,

“If the side of a square doubles, then its area increases four times”, is :

A

If the side of a square is not doubled, then its area does not increase four times.

B

If the area of a square increases four times, then its side is doubled.

C

If the area of a square increases four times, then its side is not doubled.

D

If the area of a square does not increase four times, then its side is not doubled.

Question 110: Palette 110/120

JEE Mains 2016 MCQ

Consider the following two statements :

P :     If 7 is an odd number, then 7 is divisible by 2.
Q :    If 7 is a prime number, then 7 is an odd number

If  V₁ is the truth value of the contrapositive of P and V₂ is the truth value of contrapositive of Q, then the ordered pair (V₁ , V₂) equals :

A

(T, T)

B

(T, F)

C

(F, T)

D

(F, F)

Question 111: Palette 111/120

JEE Mains 2016 MCQ

The Boolean expression

$\left( {p \wedge \sim q} \right) \vee q \vee \left( { \sim p \wedge q} \right)$ is equivalent to :

A

${ \sim p \wedge q}$

B

${p \wedge q}$

C

$p \vee q$

D

$p \vee \sim q$

Question 112: Palette 112/120

JEE Mains 2015 MCQ

The negation of $ \sim s \vee \left( { \sim r \wedge s} \right)$ is equivalent to :

A

$s \vee \left( {r \vee \sim s} \right)$

B

$s \wedge r$

C

$s \wedge \sim r$

D

$s \wedge \left( {r \wedge \sim s} \right)$

Question 113: Palette 113/120

JEE Mains 2014 MCQ

The statement $ \sim \left( {p \leftrightarrow \sim q} \right)$ is :

A

equivalent to ${ \sim p \leftrightarrow q}$

B

a tautology

C

a fallacy

D

equivalent to ${p \leftrightarrow q}$

Question 114: Palette 114/120

JEE Mains 2013 MCQ

Consider :
Statement − I : $\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \wedge q} \right)$ is a fallacy.
Statement − II :$\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right)$ is a tautology.

A

Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I

B

Statement -I is True; Statement -II is False.

C

Statement -I is False; Statement -II is True

D

Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I

Question 115: Palette 115/120

JEE Mains 2012 MCQ

The negation of the statement “If I become a teacher, then I will open a school” is :

A

I will become a teacher and I will not open a school

B

Either I will not become a teacher or I will not open a school

C

Neither I will become a teacher nor I will open a school

D

I will not become a teacher or I will open a school

Question 116: Palette 116/120

JEE Mains 2011 MCQ

Consider the following statements
P : Suman is brilliant
Q : Suman is rich
R : Suman is honest
The negation of the statement,

“Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as :

A

$ \sim \left[ {Q \leftrightarrow \left( {P \wedge \sim R} \right)} \right]$

B

$ \sim Q \leftrightarrow P \wedge R$

C

$ \sim \left( {P \wedge \sim R} \right) \leftrightarrow Q$

D

$ \sim P \wedge \left( {Q \leftrightarrow \sim R} \right)$

Question 117: Palette 117/120

JEE Mains 2010 MCQ

Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P?

A

There is no rational number x ∈ S such that x ≤ 0

B

Every rational number x ∈ S satisfies x ≤ 0

C

x ∈ S and x ≤ 0 $ \Rightarrow $ x is not rational

D

There is a rational number x ∈ S such that x ≤ 0

Question 118: Palette 118/120

JEE Mains 2009 MCQ

Statement-1 : $ \sim \left( {p \leftrightarrow \sim q} \right)$ is equivalent to ${p \leftrightarrow q}$.
Statement-2 : $ \sim \left( {p \leftrightarrow \sim q} \right)$ is a tautology.

A

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

B

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

C

Statement-1 is true, Statement-2 is false

D

Statement-1 is false, Statement-2 is true

Question 119: Palette 119/120

JEE Mains 2008 MCQ

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number iff y is a transcendental number”.

Statement –1: r is equivalent to either q or p.

Statement –2: r is equivalent to $ \sim \left( {p \leftrightarrow \sim q} \right)$

A

Statement − 1 is false, Statement − 2 is false

B

Statement −1 is false, Statement −2 is true

C

Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

D

Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1

Question 120: Palette 120/120

JEE Mains 2008 MCQ

The statement $p \to \left( {q \to p} \right)$ is equivalent to

A

$p \to \left( {p \leftrightarrow q} \right)$

B

$p \to \left( {p \to q} \right)$

C

$p \to \left( {p \vee q} \right)$

D

$p \to \left( {p \wedge q} \right)$