Method of Contradiction

Q.

What is Idempotent law?

Q.

Sameer plant $7225$ plants. So that there are as many rows as there are trees in a row. How many trees are in a row?

Q. The proposition $(p\Rightarrow \sim p)\wedge (\sim p\Rightarrow p)$ is :

1. a tautology
2. a contradiction
3. neither a tautology nor a contradiction
4. logically equivalent to $x\vee (\sim x)$

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

1. a tautology
2. a contradiction
3. logically equivalent to $x\vee (\sim x)$
4. neither a tautology nor a contradiction

Q.

Grammatical Words In Sentences Are Called ________

Q. Which among the following statements is not a contradiction

1. $[\sim p\wedge (p\vee \sim q\left)\right]\wedge q$
2. $\sim (p\wedge q)\wedge p$
3. $\left[\right(p\to q)\to p]\wedge \sim p$
4. $(\sim p\wedge \sim q)\wedge (p\wedge q)$

Q. Calculate value of root 171 using binomial approximation.

Q. If $P=a\times b\times c$
where $a,b,c$ are prime numbers and $a$ is the smallest prime number, then which of the following is correct ?

1. $P$ is always odd number
2. $P$ is always even number
3. $P$ is always prime number
4. None of these

Q. Which of the following statements is/are correct?

1. If two numbers are coprime to each other, then both the numbers must be prime.
2. Two different prime numbers are always coprime to each other.
3. Every natural number is coprime with $1.$
4. Two even numbers can never be coprime to each other.

Q.

In the method of contradiction, in order to validate a statement we need to find an exception or a contradictory case for the given statement.

1. False

2. True

Q.

Determine whether the argument used to check the validity of the following statement is correct :

p : " If $x^2$ is irrational, then x is rational "

The statement is true because the number $x^2={\pi }^2$ is irrational, therefore $x={\pi }^2$ is irrational.

Q. Counter example to the statement "All prime numbers are odd." is

1. The prime number $2$
2. None of these
3. The prime number $3$
4. Number $1$

Q. Which of the following is the correct steps to take when proving a statement using proof by contradiction?

1. 1) Assume that your statement is true.
   2) Show this is the case using definitions and theorems.
   3) State that the statement is true.
2. 1) Assume your statement is true for a certain instance.
   2) Show that it is true in more than one instance.
   3) State that your statement must be true.
3. 1) Assume your statement to be false.
   2) Proceed as you would in a direct proof.
   3) Come across a contradiction.
   4) Use the contradiction to state that your assumption of the statement being false can't be the case, so your statement must be true.
4. None of the answers are correct.

Q. The given equation $4xy-x-y=z^2$ has:

1. three positive integer solutions
2. one positive integer solutions
3. two positive integer solutions
4. no positive integer solutions

Q. Evaluate $\frac{2}{5}\times \frac{-3}{7}-\frac{3}{7}\times \frac{3}{5}-\frac{1}{14}$

Q. Prove that a positive integer $n$ is prime number, if no prime $p$ less than or equal to $\sqrt{n}$ divides $n.$

Q. To prove the statement: "Tangent of a circle is perpendicular to a radius at the point of contact.", the proof was started by "Assume tangent is not perpendicular to radius."
Above method of proof is

1. Direct method
2. Contrapositive way
3. Proof by contradiction
4. None

Q. To prove any preposition by "giving counter example" we must give at-least ______ example(s).

1. Two
2. One
3. Three
4. more than three

Q. Statement: When $x$ and $y$ are odd integers, there does not exist an odd integer $z$ such that $x+y=z$.

1. When $x$ and $y$ are odd integers, there does exist an even integer $z$ such that $x+y=z.$
2. When $x$ and $y$ are odd integers, there does not exist an odd integer $z$ such that $x+y=z$.
3. When $x$ and $y$ are odd integers, there does not exist an even integer $z$ such that $x+y=z$.
4. When $x$ and $y$ are odd integers, there does exist an odd integer $z$ such that $x+y=z$.

Q. The area of the figure formed by the intersection of lines $x=0$, $y=0$, $x=3$, $y=4$ will be

1. 4 sq. unit
2. 3 sq. unit
3. 6 sq. unit
4. 12 sq. unit

Q. Statement: There are an infinite number of prime numbers.

1. There is a finite number of prime numbers.
2. All prime numbers are less than $100$.
3. There are no prime numbers that are even.
4. There is an infinite number of primes.

Q. To prove "" All prime numbers are not odd." we showed that "$2$ is even and prime"
This method is

1. The proof by giving counter example.
2. The proof by contradiction.
3. Induction method.
4. The Direct method.

Q. Let $\overrightarrow{u}=\overrightarrow{i}+\overrightarrow{j},\overrightarrow{v}=\overrightarrow{i}-\overrightarrow{j}$ and $\overrightarrow{\omega }=\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k}.$ If $\hat{n}$ unit vector such that $\overrightarrow{u}\cdot \hat{n}=0$ and $\overrightarrow{v}\cdot \hat{n}=0$ then $|\overrightarrow{w}\cdot \hat{n}|$ is equal to

1. $1$
2. $2$
3. $3$
4. $0$

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

1. logically equivalent to $x\vee (\sim x)$
2. a contradiction
3. a tautology
4. neither a tautology nor a contradiction

Q. Mathematical models provide

1. accurate results
2. wrong results
3. estimated results
4. approximate results

Q. Fill in the blank with the most appropriate word.
I wouldn't ask for your help ________ I had no choice.

1. since
2. even if
3. despite
4. although

Q. Let P and Q be two $2\times 2$ matrices. Consider the statements i)$PQ=O\Rightarrow P=O$ or $Q=O$ or both
ii) $PQ=I_2\Rightarrow P=Q^{-1}$
iii) ${(P+Q)}^2=P^2+2PQ+Q^2$. Then

1. (i) and (iii) are false while (ii) is true
2. (i) and (ii) are false while (iii) is true
3. (ii) and (iii) are false while (i) is true
4. None