Negation

Q.

Prove that $\sqrt{3}$ is an irrational number.

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

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

Q. Consider the following statements:
P: Ramu is intelligent.
Q: Ramu is rich.
R: Ramu is not honest.
The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:

1. $\left(\right(P\wedge (\sim R))\wedge Q)\wedge \left(\right(\sim Q)\wedge ((\sim P)\vee R\left)\right)$
2. $\left(\right(P\wedge R)\wedge Q)\vee \left(\right(\sim Q)\wedge ((\sim P)\vee (\sim R)\left)\right)$
3. $\left(\right(P\wedge R)\wedge Q)\wedge \left(\right(\sim Q)\wedge ((\sim P)\vee (\sim R)\left)\right)$
4. $\left(\right(P\wedge (\sim R))\wedge Q)\vee \left(\right(\sim Q)\wedge ((\sim P)\vee R\left)\right)$

Q.

Consider the following statements

P : Suman is brilliant

Q: Suman is rich

R: Suman is honest

The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as

1. $\sim (Q\leftrightarrow (P\wedge \sim R\left)\right)$
2. $\sim Q\leftrightarrow \sim P\wedge R$
3. $\sim (P\wedge \sim R)\leftrightarrow Q$
4. $\sim P\wedge (Q\leftrightarrow \sim R)$

Q. Negation of the statement $(p\vee r)\Rightarrow (q\vee r)$ is

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

Q.

The negation of the statement: “If I become a teacher, then I will open a school”, is

1. I will become a teacher and I will not open a school

2. Either I will not become a teacher or I will not open a school

3. Neither I will become a teacher nor I will open a school

4. I will not become a teacher or I will open a school

Q. Negation of the statement : ' $\sqrt{5}$ is an integer or $5$ is irrational ' is :

1. $\sqrt{5}$ is irrational or $5$ is an integer
2. $\sqrt{5}$ is not an integer or $5$ is not irrational
3. $\sqrt{5}$ is an integer and $5$ is irrational
4. $\sqrt{5}$ is not an integer and $5$ is not irrational

Q.

The conditional $(p˄q)\Rightarrow p$is

1. A tautology

2. A fallacy i.e., contradiction

3. Neither tautology nor fallacy

4. None of these

Q.

Negation of “Paris is in France and London is in England” is

1. Paris is in England and London is in France

2. Paris is not in France or London is not in England

3. Paris is in England or London is in France

4. None of these

Q.

Using the extended euclidean algorithm, find the multiplicative inverse of $12$in $Z26$

Q. The negation of the statement: It is raining and it is cool is

1. It is not raining and it is cool.
2. It is raining and it is not cool.
3. Neither it is raining nor it is cool.
4. It is not raining or it is not cool.

Q.

Which country is called the "Land of Morning Calm"?

Q. Let $S$ be a non-empty subset of $R$. Consider the following statement :
$P:$ There is a rational number $x\in S$ such that $x>0$.
Which of the following statements is the negation of the statement $P$ ?

1. $x\in S$ and $x\le 0\Rightarrow x$ is not rational.
2. There is a rational number $x\in S$ such that $x\le 0$
3. There is no rational number $x\in S$ such that $x\le 0$
4. Every rational number $x\in S$ satisfies $x\le 0$

Q.

If$p:$ $2$ plus $3$ is five and$q:$ Delhi is the capital of India $<$are two statements, then the statement “Delhi is the capital of India and it is not that $2$ plus $3$ is five” is

1. $~p\vee q$

2. $~p\wedge q$

3. $p\wedge ~q$

4. $p\vee ~q$

5. $~p\wedge ~q$

Q. The negation of the Boolean expression $\sim s\vee (\sim r\wedge s)$ is equivalent to

1. $s\vee r$
2. $s\wedge r$
3. $r$
4. $\sim s\wedge \sim r$

Q. If A=\begin{bmatrix}0&i -i&0\end{bmatrix} , then prove that A^{40}= \begin{bmatrix}1&0 0&1\end{bmatrix

Q.

$0$ is not identity for subtraction.

1. True
2. False

Q. The negation of the statement: ''It is raining and it is cool'' is

1. It is not raining and it is cool.
2. It is raining and it is not cool.
3. Neither it is raining nor it is cool.
4. It is not raining or it is not cool.

Q. The negation of $\sim s\vee (\sim r\wedge s)$ is equivalent to

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

Q.

Write the opposites of

Q.

The negation of the conditional statement ‘If it rains, I shall go to school’ is___.

1. It rains and I shall go to school

2. It does not rain and I shall go to school

3. It does not rain and I shall not go to school

4. It rains and I shall not go to school

Q. The Boolean expression $\sim (p\Rightarrow (\sim q\left)\right)$ is equivalent to :

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

Q. The negation of the statement ''If $X$ becomes the president of party $Y,$ then $X$ will be our next prime minister'' is

1. $X$ becomes the president of party $Y$ and $X$ will not be our next prime minister
2. $X$ becomes the president of party $Y$ or $X$ will not be our next prime minister
3. If $X$ becomes the president of party $Y,$ then $X$ will not be our next prime minister
4. If $X$ does not become the president of party $Y,$ then $X$ will not be our next prime minister

Q.

Are the following pairs of statements are negation of each other.

(i) The number x is not a rational number.

The number x is not an irrational number.

(ii) The number x is not a rational number.

The number x is an irrational number.

Q.

How to write statements in solutions of permutation based questions?

Q.

Negate each of the following statements :

(i) All the students completed their homework.

(ii) There exists a number which is equal to its square.

Q. Which of the following is not a negation for the statement ''for every $x\in N,x+3<10$''

1. There exists a number $x$ such that $x\in N,x+3\ge 10$.
2. Not for every $x\in N,x+3<10$.
3. It is false that for every $x\in N,x+3<10$.
4. for every $x\in N,x+3>10$.

Q.

Which of the following numbers is divisible by $99$

1. $913462$

2. $114345$

3. $135792$

4. $3572406$

Q. A fraction becomes equal to $\frac{4}{5}$ if 1 is added to both numerator and denominator. If, however, 5 is subtracted from both numerator and denominator, the fraction becomes equal to $\frac{1}{2}$. What is the fraction?

Q. The negation of the statement $(p\wedge q)\to (\sim p\vee r)$ is

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