1. ⇒ (MHT CET 2023 11th May Evening Shift )

If truth value of logical statement $(p \leftrightarrow\sim q) \to (\sim p \land q)$ is false, then the truth values of $p$ and $q$ are respectively

A. $F, T$

B. $T, T$

C. $T, F$

D. $F, F$

Correct answer option is (C)

As $(p \leftrightarrow\sim q) \to (\sim p \land q)$ is false, we get

$p \leftrightarrow\sim q \equiv T$ and $\sim p \land q \equiv F$

$\sim p \land q \equiv F$

$\Rightarrow\sim p \equiv F$ and $q \equiv F$

$\Rightarrow p \equiv T$ and $q \equiv F$

2. ⇒ (MHT CET 2023 10th May Evening Shift )

If $p$ and $q$ are true statements and $r$ and $s$ are false statements, then the truth values of the statement patterns $(p \land q) \lor r$ and $(p \lor s) \leftrightarrow (q \land r)$ are respectively

A. F, T

B. $T, T$

C. F, F

D. $T, F$

Correct answer option is (D)

$\begin{array}{ll} (p \land q) \lor r & (p \lor s) \leftrightarrow (q \land r) \\ \equiv (T \land T) \lor F & \equiv (T \lor F) \leftrightarrow (T \land F) \\ \equiv T \lor F & \equiv T \leftrightarrow F \\ \equiv T & \equiv F \end{array}$

3. ⇒ (MHT CET 2023 9th May Morning Shift )

If truth values of statements $p, q$ are true, and $r$ , $s$ are false, then the truth values of the following statement patterns are respectively

$\begin{aligned} a :\sim (p \land \sim r) \lor (\sim q \lor s) \\ b : (\sim q \land \sim r) \leftrightarrow (p \lor s) \\ c : (\sim p \lor q) \to (r \land \sim s) \end{aligned}$

A. $T, F, F$

B. F, F, F

C. F, T, T

D. $T, F, T$

Correct answer option is (B)

$\begin{aligned} a. & \sim (p \land \sim r) \lor (\sim q \lor s) \\ \equiv\sim (T \land \sim F) \lor (\sim T \lor F) \\ \equiv (F \lor F) \lor (F \lor F) \\ \equiv F \lor F \\ \equiv F \end{aligned}$

$\begin{aligned} b. & (\sim q \land \sim r) \leftrightarrow (p \lor s) \\ \equiv (\sim T \land \sim F) \leftrightarrow (T \lor F) \\ \equiv F \leftrightarrow T \\ \equiv F \end{aligned}$

$\begin{aligned} c. & (\sim p \lor q) \to (r \land \sim s) \\ \equiv (\sim T \lor T) \to (F \land \sim F) \\ \equiv (F \lor T) \to (F \land T) \\ \equiv T \to F \\ \equiv F \end{aligned}$

4. ⇒ (MHT CET 2021 21th September Evening Shift )

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

A. T, F

B. F, F

C. F, T

D. T, T

Correct answer option is (A)

$p \to (\sim p \lor q) \equiv F$

We know that $T \to F \equiv F$

$∴ p \equiv T and (\sim p \lor q) \equiv F .$

We know that $F \lor F \equiv F$

$∴\sim p \equiv F and q \equiv F$

Thus $p, q$ are $T, F$ respectively.

5. ⇒ (MHT CET 2021 21th September Morning Shift )

If $p, q$ are true statements and $r$ is false statement, then which of the following is correct.

A. $(p \lor q) \lor r$ has truth value $F$ .

B. $(p \to r) \to q$ has truth value $F$ .

C. $(p \land q) \to r$ has truth value $T$ .

D. $(p \leftrightarrow q) \to r$ has truth value $F$ .

Correct answer option is (D)

We will go by options

(A) $(p \lor q) \lor r \equiv (T \lor T) \lor F \equiv T \lor F \equiv T$

(B) $(p \land q) \to r \equiv (T \land T) \to F \equiv T \to F \equiv F$

(C) $(p \to r) \to q \equiv (T \to F) \to T \equiv F \to T \equiv T$

(D) $(p \leftrightarrow q) \to r \equiv (T \leftrightarrow T) \to F \equiv T \to F \equiv F$

⇒ Mathematical Logic