Logical Connectives

Q. Which of the following propositions is a tautology

1. $p\vee (q\to p)$
2. $p\vee (p\to q)$
3. Both (a) and (c)
4. $(p\wedge q)\to q$

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

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

Q. 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

1. $B\to \left(\right(\sim A)\vee (\sim C\left)\right)$
2. $(\sim B)\wedge (A\wedge C)$
3. $B\to (A\vee C)$
4. $B\to (A\wedge C)$

Q. In the question below, a sentence has been given in direct/indirect speech. From the given alternatives, choose the one which best expresses the given sentence in indirect/direct speech.

I told him that he was not working hard.

1. I said to him, "You are not working hard."
2. I told to him, "You are not working hard."
3. I said, "You are not working hard."
4. I said to him, "He is not working hard."

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

1. $FFT$
2. $FTF$
3. $TFF$
4. $TFT$

Q.

Abstract Noun of Intelligent

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. $r\to (p\wedge \sim q)$
4. $\sim p\to (p\wedge r)$

Q.

What Is the Duration of Normal Hockey Match

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. The proposition $\sim (p\iff q)$ is equivalent to

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

Q.

What is the abstract noun for clever?

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

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. 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:

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

Q. Let the operations $\ast ,\odot \in \{\wedge ,\vee \}$. If $(p\ast q)\odot (p\odot \sim q)$ is a tautology, then the ordered pair $(\ast ,\odot )$ is

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

Q. The contrapositive of $2x+3y=9\Rightarrow x\ne 4$ is

1. $x=4\Rightarrow 2x+3y\ne 9$
2. $x=4\Rightarrow 2x+3y=9$
3. $x\ne 4\Rightarrow 2x+3y\ne 9$
4. $x\ne 4\Rightarrow 2x+3y=9$

Q.

State whether the given statement is True(T) or False (F).

Two adjacent angles are always complementary.

1. True
2. False

Q.

Write the given number with appropriate signs:

$100m$ below sea level.

Q.

What Is Abstract Noun Of Clever

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. Negation of the statement $\sim p\to (q\vee r)$ is

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

Q. In the question below, a sentence has been given in direct/indirect speech. From the given alternatives, choose the one which best expresses the given sentence in indirect/direct speech.
She said to me, ''What time is your flight tomorrow?''

1. She told me what time was my flight the next day.
2. She told me what time my flight will be the next day.
3. She asked me what time my flight was the next day.
4. She asked me that what time will be my flight tomorrow.

Q.

How many months does coordinate Geometry takes to cover in class 11 (approx) ?

Q. State and prove binomial theorem

Q. The only statement among the following that is a tautology is

1. $b\to [a\wedge (a\to b\left)\right]$
2. $a\vee (a\wedge b)$
3. $a\wedge (a\vee b)$
4. $[a\wedge (a\to b\left)\right]\to b$

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. 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.

Write opposites of the following:

$100\mathrm{m}$ above sea level

Q. The statement $p\wedge (q\iff r)$ is a

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