Tautology

Q. The statement $\left[\right(p\Rightarrow q)\wedge (q\Rightarrow r\left)\right]\Rightarrow (p\Rightarrow r)$ is

1. a tautology
2. a contradiction
3. neither a tautology nor contradiction
4. logically equivalent to $(p\wedge q)\vee r$

Q. If $q$ is false and $p\wedge q\leftrightarrow r$ is true, then which one of the following statements is a tautology?

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

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

1. A tautology
2. A contradiction
3. Logically equivalent to $p\wedge q$
4. Niether tautology nor contradiction

Q. Which one of the following statements is not a tautology ?

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

Q.

Given the following two statements

$\left(S_1\right):(q\vee p)\to (p\leftrightarrow ~q)$ is a tautology.

$\left(S_2\right):~q\wedge (~p\leftrightarrow q)$ is a fallacy. Then:

1. only $\left(S_1\right)$ is correct.

2. both $\left(S_1\right)$ and $\left(S_2\right)$ are correct.

3. only $\left(S_2\right)$ is correct

4. both $\left(S_1\right)$ and $\left(S_2\right)$ are not correct.

Q. Let $\Delta ,\nabla \in \{\wedge ,\vee \}$ be such that $p\nabla q\Rightarrow \left(\right(p\Delta q\left)\nabla r\right)$ is a tautology. Then $\left(p\nabla q\right)\Delta r$ is logically equivalent to:

1. $(p\wedge r)\Delta q$
2. $\left(p\nabla r\right)\wedge q$
3. $\left(p\Delta r\right)\vee q$
4. $\left(p\Delta r\right)\wedge q$

Q.

Which of the following is not a tautology ?

1. $p\vee \sim p$

2. $(p\Rightarrow q)\Rightarrow (\sim q\Rightarrow \sim p)$

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

4. $p\wedge (q\vee \sim q)$

Q. Consider the following three statements:
$P:5$ is a prime number
$Q:7$ is a factor of 192
$R:\text{L.C.M. of}5\text{and}7\text{is}35$
Then the truth value of which one of the following statements is true?

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

Q. Given the following two statements:
$\left(S_1\right):(q\vee p)\to (p\leftrightarrow \sim q)$ is a tautology.
$\left(S_2\right):\sim q\wedge (\sim p\leftrightarrow q)$ is a fallacy. Then

1. Only $\left(S_1\right)$ is correct
2. Both $\left(S_1\right)$ and $\left(S_2\right)$ are correct
3. Only $\left(S_2\right)$ is correct
4. Both $\left(S_1\right)$ and $\left(S_2\right)$ are not correct

Q.

The statement $(p\Rightarrow q)\iff (~p\wedge q)$ is a

1. Tautology

2. contradiction

3. Neither A nor B

4. none of these

Q.

Consider the following statements:

Statement I:$(p\wedge ~q)\wedge (~p\wedge q)$ is a fallacy.

Statement II: $(p\to q)\leftrightarrow (~q\to ~p)$ is a tautology.

1. Statement I is correct, Statement II is correct; Statement II is a correct explanation for Statement I

2. Statement I is correct; Statement II is correct; Statement II is not a correct explanation for Statement I

3. Statement I is correct, Statement II is incorrect

4. Statement I is incorrect, and Statement II is correct

Q. For the statements $p$ and $q,$ consider the following compound statements :
(a) $(\sim q\wedge (p\to q\left)\right)\to \sim p$
(b) $\left(\right(p\vee q)\wedge \sim p)\to q$
Then which of the following ststements is correct ?

1. (a) is a tautology but not (b).
2. (a) and (b) both are not tautologies.
3. (a) and (b) both are tautologies.
4. (b) is a tautology but not (a).

Q. The following statement $(p\to q)\to \left[\right(\sim p\to q)\to q]$ is:

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

Q. Which one of the following Boolean expressions is a tautology ?

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

Q.

Choose the correct option.

Statement $1:(p\wedge \sim q)\wedge (\sim p\wedge q)$ is a fallacy.

Statement $2:(p\Rightarrow q)\iff (\sim q\Rightarrow \sim p)$ is a tautology.

1. STATEMENT $1$ is FALSE and STATEMENT $2$ is TRUE.
2. Both the statements are TRUE and STATEMENT $2$ is the correct explanation of STATEMENT $1.$
3. Both the statements are TRUE but STATEMENT $2$ is NOT the correct explanation of STATEMENT $1.$
4. STATEMENT $1$ is TRUE and STATEMENT $2$ is FALSE.

Q. Which among the following statement is incorrect

1. $p\wedge (\sim p)$ is a contradiction
2. $(p\Rightarrow q)\iff (\sim q\Rightarrow \sim p)$ is a contradiction
3. $\sim (\sim p)\iff p$ is a tautology
4. $p\vee (\sim p)$ is a tautology

Q. Which of the following statements is a tautology?

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

Q. Which of the following is a tautology

1. $a\vee b\to b\wedge c$
2. $a\wedge b\to b\vee c$
3. $a\vee b\to (b\to c)$
4. Both $\left(a\right)$ and $\left(c\right)$ are true

Q. The logical statement $(p\to q)\to ((\sim p\to q)\to q)$ is

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

Q. Let $p,q,r$ be three statements. Then $(p\to (q\to r\left)\right)⟹\left(\right(p\wedge q)\to r)$ is

1. neither tautology nor fallacy
2. tautology
3. fallacy
4. contradiction

Q. Is $(p\to q)\vee (q\to p)$ a tautology ?

1. True
2. False

Q. Assertion :The contrapositive and converse of the statement: "If $x$ is a prime number, then $x$ is odd" are:
Contrapositive: "If $x$ is not odd, then $x$ is not a prime number.
Converse: If $x$ is odd, then $x$ is a prime number. Reason: If statement $p$ implies $q$, then its contrapositive is $\sim q$ implies $\sim p$, and its converse is $q$ implies $p$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
3. Assertion is correct but Reason is incorrect.
4. Both Assertion and Reason are incorrect.

Q. $p\wedge q$ is logically equivalent to

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

Q. For any two statements $p$ and $q,$ the expression $\sim (p\vee q)\vee (\sim p\wedge q)$ is logically equivalent to

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

Q.

The statement $(p\to q)\to \left[\right(\sim p\to q)\to q]$ is

1. a tautology

2. equivalent to $\sim p\to q$

3. equivalent to $p\to \sim q$

4. a fallacy

Q. In the following questions some parts of the sentence have been jumbled up. You are required to rearrange these parts which are labelled as P, Q, R and S to produce the correct and meaningful sentence. Choose the sequence and mark the correct answer.
(1) Before presenting a paper it is advisable to choose a subject of your interest.
(P) When presenting your paper, speak distinctly and pleasantly.
(Q) Next, 'read up' about it, before you write the paper.
(R) Then prior to writing make a clear plan to ensure orderly presentation.
(S) While reading, you should make notes
(6) In this way, the interest of the listeners is enhanced and you are appreciated.

1. RSQP
2. PQRS
3. QSRP
4. SQRP.

Q. Dual of: $p\wedge [\sim q\vee (p\wedge q)\vee \sim r]$

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

Q. If $p$ and $q$ are prime numbers satisfying the condition $p^2-2q^2=1$, then the value of $p^2+2q^2$ is

1. $17$
2. $16$
3. $5$
4. $15$

Q. $[p\wedge (\sim q\left)\right]\wedge \left[\right(\sim p)\vee q]$

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

Q.

Which of the following is not a tautology ?

1. $p\vee \sim p$

2. $(p\Rightarrow q)\Rightarrow (\sim q\Rightarrow \sim p)$

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

4. $p\wedge (q\vee \sim q)$