Implication

Q.

Which of the following are equivalent statements to the implication $p\to q$.

1. $\sim q\to \sim p$

2. $p$ only if $q$

3. $p$ is a sufficient condition for $q$

4. $q$ is a necessary condition for $p$

Q. Let p : Kiran passed the examination,
q : Kiran is sad
The symbolic form of a statement "It is not true that Kiran passed therefore he is sad' is

1. $(\sim p\to q)$
2. $(p\to q)$
3. $\sim (p\to \sim q)$
4. $\sim (p\leftrightarrow q)$

Q. Which of the following is logically equivalent to $\sim (\sim p\Rightarrow q)$

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

Q. The statement $p\to (q\to p)$ is equivalent to

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

Q. Are these statements logically equivalent?
Statement 1: If it does not rain, I will play cricket.
Statement 2: I will not play cricket if it rains.

1. Yes
2. No

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

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

Q.

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

1. T, F, T

2. F, T, T

3. F, F, T

4. T, F, F

Q. Let $p$ and $q$ be any two logical statements and $r$ be the statement $p\to (\sim p\vee q).$ If $r$ has the truth value $\text{F},$ then the truth values of $p$ and $q$ are, respectively

1. $\text{F, F}$
2. $\text{T, T}$
3. $\text{F, T}$
4. $\text{T, F}$

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

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

Q. If the statement $(p\to q)\to (q\to r)$ is false, then truth values of statement $p,q,r$ respectively, can be

1. $\text{F T F}$
2. $\text{T F T}$
3. $\text{T F F}$
4. $\text{F T T}$