Logical Connectives

Q. The following statement $(p\to q)\to \left[\right(\sim p\to q)\to q]$ is:

1. a tautology
2. equivalent to $\sim p\to q$
3. equivalent to $p\to \sim q$
4. a fallacy

Q. Which of the following biconditional statements are true?

1. $2\ne 3\leftrightarrow 2>3\mathrm{or}2<3$
2. $2=3\leftrightarrow 3=4$
3. $a<3\leftrightarrow a>3$
4. $a=3\leftrightarrow a\le 3\mathrm{and}a\ge 3$

Q. If a then b and if b then c $⟹$ If a then c.

1. False
2. True

Q. The boolean expression $\left(\right(p\wedge q)\vee (p\vee \sim q\left)\right)\wedge (\sim p\wedge \sim \text{q})$ is equivalent to:

1. $p\wedge (\sim q)$
2. $(\sim p)\wedge (\sim q)$
3. $p\vee (\sim q)$
4. $p\wedge q$

Q. The logical statement $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ is equivalent to

1. $\sim p$
2. $p$
3. $q$
4. $\sim q$

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

1. $(\vee ,\vee )$
2. $(\vee ,\wedge )$
3. $(\wedge ,\wedge )$
4. $(\wedge ,\vee )$

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

1. $T,T,F$
2. $T,F,T$
3. $T,F,F$
4. $F,T,T$

Q. The inverse of the proposition $(p\wedge \sim q)\to r$ is

1. $\sim r\to (\sim p\vee q)$
2. $(\sim p\vee q)\to \sim r$
3. $\sim p\to (p\wedge r)$
4. $r\to (p\wedge \sim q)$

Q.

The dual of statement $p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)$ is

1. $p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)$

2. $p\vee (q\wedge r)\equiv (p\wedge q)\wedge r$

3. $p\wedge (q\wedge r)\equiv (p\wedge q)\wedge r$

4. $p\vee (q\vee r)=(p\wedge q)\wedge r$

Q. Let $p,q,r$ denote the arbitary statements then the logical equivalance of the statement $p\Rightarrow (q\vee r)$ is

1. $(p\vee q)\Rightarrow r$
2. $(p\Rightarrow \sim q)\wedge (p\Rightarrow r)$
3. $(p\Rightarrow q)\wedge (p\Rightarrow \sim r)$
4. $(p\Rightarrow q)\vee (p\Rightarrow r)$