And

Q.

$(p\wedge \sim q)\wedge (\sim p\wedge q)$ is ___.

1. A tautology

2. A fallacy

3. Both a tautology and a fallacy

4. Neither a tautology nor a fallacy

Q. OR is a logical disjunction and AND is a logical conjunction.

1. False
2. True

Q. Consider the following three statements:
$p:4$ and $7$ are coprimes.
$q:$ Every function on a set $A$ is a relation on $A$.
$r:6$ divides $3$.
Then the truth value of which of the following Boolean expressions is false?

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

Q. Let
$p:2$ is a prime number
$q:\cos {30}^{\circ }=\frac{1}{2}$
$r:{\sec }^{2}x+{\tan }^{2}x=1$
$s:\sqrt{7}$ is an irrational number
$u:{\pi }^2$ is greater than $10$

Then which among the following statements are valid

1. $p\vee q,q\vee s,r\vee s,s\vee u$
2. $p\wedge q,q\wedge s,r\wedge s,s\wedge u$
3. $p\vee q,q\vee r,r\vee s,s\vee u$
4. $p\vee q,q\wedge r,r\vee s,s\wedge u$

Q. The statement $(p\vee q)\wedge (\sim p\wedge \sim q)$ is

1. a contradiction
2. a contingency
3. logically equivalent to $x\vee (\sim x)$
4. a tautology

Q. Let $p:2$ is a prime number
$q:\cos {30}^{\circ }=\frac{1}{2}$
$r:{\sec }^{2}x+{\tan }^{2}x=1$
$s:\sqrt{7}$ is an irrational number
$u:{\pi }^2$ is greater than $10$

The statement which are all false is

1. $p\vee q,r\vee s,r\vee u$
2. $p\wedge q,q\vee r,p\vee u$
3. $p\wedge q,q\wedge s,s\wedge u$
4. $p\wedge q,q\vee r,r\vee s$

Q. The truth value of "$20+10=2$" and
"$20\times 10=200$" is

1. $F$
2. $T$
3. $T$ or $F$ based on sub statements truth value
4. cannot be determined

Q. If $p:x$ is odd; $q:x^2$ is odd. Then "$x$ is odd and $x^2$ is not odd", is represented as

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

Q. Truth value for the statement "$5+7=10$ and $4+3=7$" is

1. Depends upon the truth values of sub statement.
2. $5+7=10⟺4+3=7$ can not be determined.
3. $T$
4. $F$

Q. The disjunction statement that can be formed from statements, " It is raining"; "The sun is shining", is

1. It is raining and the sun is shining
2. It is raining or the sun is shining
3. It is raining and the sun is not shining
4. It is not raining or the sun is not shining