Validation of Statement

Q. The statement $\sim (p\leftrightarrow \sim q)$ is:

1. equivalent to $p\leftrightarrow q$
2. equivalent to $\sim p\leftrightarrow q$
3. a tautology
4. a fallacy

Q. Consider the two statements:
$\left(S1\right):(p\to q)\vee (\sim q\to p)$ is a tautology.
$\left(S2\right):(p\wedge \sim q)\wedge (\sim p\vee q)$ is a fallacy.
Then

1. only $\left(S1\right)$ is true.
2. only $\left(S2\right)$ is true.
3. both $\left(S1\right)$ and $\left(S2\right)$ are true.
4. both $\left(S1\right)$ and $\left(S2\right)$ are false.

Q. Let$p,q,r$ be three logical statements. Consider the compound statements
$S_1:\left(\right(\sim p)\vee q)\vee \left(\right(\sim p)\vee r)$ and
$S_2:p\to (q\vee r)$
Then, which of the following is NOT true?

1. If $S_2$ is False, then $S_1$ is True

2. If $S_2$ is True, then $S_1$ is True

3. If $S_2$ is False, then $S_1$ is False

4. If $S_1$ is False, then $S_2$ is False

Q. Which of the following Boolean expressions is not a tautology:

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

Q.

Which of the following statements is a tautology?

1. $(~q\wedge p)\wedge q$

2. $(~q\wedge p)\wedge (p\wedge ~p)$

3. $(~q\wedge p)\vee (p\vee ~p)$

4. $\left(p\wedge q\right)\wedge \left(~\left(p\wedge q\right)\right)$

Q.

In a triangle $\left(1-\frac{r_1}{r_2}\right)\left(1-\frac{r_1}{r_3}\right)=2$, then the triangle is

1. Equilateral.

2. Right angled.

3. Isosceles.

4. None of these.

Q. Which of the following is (are) not rational number(s)?

1. $0.2\overline{3}$
2. $\sqrt{7}$
3. $\sqrt{3}$
4. $0.\overline{567}$

Q. The truthfulness or falsity of a statement is called its

1. negation
2. disjunction
3. truth value
4. conjunction

Q. Which among the following is (are) rational number(s) ?

1. $\frac{10}{3}$
2. $0.99099$
3. $\sqrt{5}$
4. $\pi$

Q.

If p and q are simple statements, $p\iff \sim q$ is true when

1. p is true and q is true

2. Both p and q are false

3. p is false and q is true

4. none of these

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.

Which Of These Is Not A Part Of Letter?

1. Status

2. Signature

3. Greeting

4. Date

Q.

Show that the following statement is true

"The integer n is even if and only if $n^2$ is even"

Q.

Determine whether each of the following sentences is a proposition: If the sentence is a proposition, then write its negation:

1. Every prime number is even.
2. There is a number that is larger than $17$.

Q. Answer, whether the following statements are true or false. Give reasons.

If $A=\text{{}x:x\text{is an even prime number}}$, then set $A$ is empty.

Q.

If the angles $A<B<C$ of a $∆ABC$ are in A. P. then

1. $c^2=a^2+b^2–a{b}^2$

2. $c^2=a^2+b^2$

3. $a^2=b^2+c^2–bc$

4. $b^2=a^2+c^2–a{c}^2$

Q. If $p\to (p\wedge \sim q)$ is false. Then the truth values of $p$ and $q$ are respectively

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

Q.

If $(p\wedge \sim r)\Rightarrow (q\wedge r)$ is false and q and r are both false, then p is ___.

1. Cannot conclude from given the information

2. False

3. True

4. May be true or false

Q. The Boolean expression $\sim (p\vee q)\vee (\sim p\wedge q)$ is equivalent to

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

Q. Let (i) $(p\vee q)\vee (p\vee \sim q)$,
(ii) $(p\wedge q)\wedge (p\vee \sim q)$,
(iii) $(p\vee q)\wedge (p\vee \sim q)$,
(iv) $(p\vee q)\vee (p\wedge \sim q)$
which one is tautology

1. $\left(i\right)$
2. $\left(ii\right)$
3. $\left(iii\right)$
4. $\left(iv\right)$

Q.

Fill in the blanks using sign $>,<\text{or}=$

$\text{XXX \_\_\_\_}\text{XXX}$

Q. $\sim (p\vee q)\vee (\sim p\wedge q)$ is logically equivalent to

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

Q. If $A=\{x:x=4k+1,k\in W\text{and}k\le 12\}$ and $B=\{x:x=5^k,k\in W\text{and}k\le 6\}$, then $n\left(A\Delta B\right)$ is

1. $17$
2. $14$
3. $20$
4. $11$

Q. Which of the following is true?

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

Q. What is the nature of the curve $x^2+y^2-25=0$

Q.

Which of the following is a tautology?

1. $(\wedge (⟶\left)\right)⟶$

2. $\wedge (\vee )$

3. $(\to (\wedge (\to ))$

4. $\vee (\wedge )$

Q.

Sivaang made the statement "a prime number is always odd”. Dylan replied "2 is a prime number and its an even number”. By which method did Dylan validate/disprove Sivaang's statement?

1. Direct

2. Contradiction

3. Contrapositive

4. None of these

Q.

Which of the following statements are true and which are false ? In each case give a valid reason for saying so

(i) p : Each radius of a circle chord of the circle.

(ii) q : The centre of a circle bisects each chord of the circle.

(iii) r : Circle is a particular case of am ellipse.

(iv) s If x and y are integers such that x > y, then -x < -y .

(v) $t:\sqrt{11}$ is a rational number.

Q.

Prove that $6+\sqrt{2}$ is irrational:

Q.

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

1. T, F, F

2. F, F, F

3. F, T, T

4. T, T, T