Chapter 13

The negation of the Boolean expression \(p \vee(\sim p \wedge q)\) is equivalent to: \(\quad\) (a) \(p \wedge \sim q\) (b) \(\sim p \wedge \sim q\) (c) \(\sim p \vee \sim q\) (d) \(-p \vee q\)

The negation of the Boolean expression \(x \leftrightarrow \sim y\) is equivalent to: \(\quad\) (a) \((x \wedge y) \vee(\sim x \wedge-y)\) (b) \((x \wedge y) \wedge(\sim x \vee \sim y)\) (c) \((x \wedge \sim y) \vee(\sim x \wedge y)\) (d) \((\sim x \wedge y) \vee(\sim x \wedge \sim y)\)

Given the following two statements : \(\left(S_{1}\right):(q \vee p) \rightarrow(p \leftrightarrow \sim q)\) is a tautology. \(\left(S_{2}\right): \sim q \wedge(\sim p \leftrightarrow q)\) is a fallacy. Then : (a) both \(\left(S_{1}\right)\) and \(\left(S_{2}\right)\) are correct (b) only \(\left(S_{1}\right)\) is correct (c) only \(\left(S_{2}\right)\) is correct (d) both \(\left(S_{1}\right)\) and \(\left(S_{2}\right)\) are not correct

The proposition \(p \rightarrow \sim(p \wedge \sim q)\) is equivalent to : (a) \(q\) (b) \((\sim p) \vee q\) (c) \((\sim p) \wedge q\) (d) \((\sim p) \vee(\sim q)\)

Let \(p, q, r\) be three statements such that the truth value of \((p \wedge q) \rightarrow(\sim q \vee r)\) is \(\mathrm{F}\). Then the truth values of \(p, q, r\) are respectively: (a) \(\mathrm{T}, \mathrm{F}, \mathrm{T}\) (b) \(\mathrm{T}, \mathrm{T}, \mathrm{T}\) (c) \(\mathrm{F}, \mathrm{T}, \mathrm{F}\) (d) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\)

If \(p \rightarrow(p \wedge \sim q)\) is false, then the truth values of \(p\) and \(q\) are respectively: (a) \(\mathrm{F}_{1} \mathrm{~F}\) (b) \(\mathrm{T}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{T}\) (d) \(\mathrm{F}, \mathrm{T}\)

Which one of the following is a tautology? (a) \((p \wedge(p \rightarrow q)) \rightarrow q\) (b) \(q \rightarrow(p \wedge(p \rightarrow q))\) (c) \(p \wedge(p \vee q)\) (d) \(p \vee(p \wedge q)\)

Which of the following statements is a tautology? (a) \(p \vee(\sim q) \rightarrow p \wedge q\) (b) \(\sim(p \wedge \sim q) \rightarrow p \vee q\) (c) \(\sim(p \vee \sim q) \rightarrow p \wedge q\) (d) \(\sim(p \vee \sim q) \rightarrow p \vee q\)

The logical statement \((\mathrm{p} \Rightarrow \mathrm{q}) \wedge(\mathrm{q} \Rightarrow \sim \mathrm{p})\) is equivalent to: \(\quad\) (a) \(p\) (b) \(q\) (c) \(\sim p\) (d) \(\sim q\)

If the truth value of the statement \(p \rightarrow(\sim q \vee r)\) is false (F), then the truth values of the statements \(p, q, r\) are respectively. \(\quad\) (a) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\) (b) \(\mathrm{T}, \mathrm{F}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{F}, \mathrm{T}\) (d) \(\mathrm{F}, \mathrm{T}, \mathrm{T}\)

The Boolean expression \(\sim(p \Rightarrow(\sim q))\) is equivalent to : (a) \(p \wedge \bar{q}\) (b) \(q \Rightarrow \sim p\) (c) \(p \vee q\) (d) \((\sim p) \Rightarrow q\)

Which one of the following Boolean expressions is a tautology ? (a) \((\mathrm{p} \wedge q) \vee(\mathrm{p} \wedge \sim \mathrm{q})\) (b) \((p \vee q) \vee(p \vee \sim q)\) (c) \((p \vee q) \wedge(p \vee \sim q)\) (d) \((p \vee q) \wedge(\sim p \vee \sim q)\)

If \(p \Rightarrow(q \vee r)\) is false, then the truth values of \(p, q, r\) are respectively: (a) \(\mathrm{F}, \mathrm{T}, \mathrm{T}\) (b) \(\mathrm{T}, \mathrm{F}, \mathrm{F}\) (c) \(\mathrm{T}, \mathrm{T}, \mathrm{F}\) (d) \(\mathrm{F}, \mathrm{F}, \mathrm{F}\)

Which one of the following statements is not a tautology? (a) \((p \vee q) \rightarrow(p \vee(\sim q))\) (b) \((p \wedge q) \rightarrow(\sim p) \vee q\) (c) \(p \rightarrow(p \vee q)\) (d) \((p \wedge q) \rightarrow p\)

The Boolean expression \(((\mathrm{p} \wedge \mathrm{q}) \vee(\mathrm{p} \vee \sim \mathrm{q})) \wedge(\sim \mathrm{p} \wedge \sim \mathrm{q})\) is equivalent to (a) \(\mathrm{p} \wedge \mathrm{q}\) (b) \(\mathrm{p} \wedge(-q)\) (c) \((\sim p) \wedge(-q)\) (d) \(\mathrm{pv}(-\mathrm{q})\)

The expression \(-(-\mathrm{p} \rightarrow \mathrm{q})\) is logically equivalent to: (a) \(-\mathrm{p} \wedge-\mathrm{q}\) (b) \(\mathrm{p} \wedge-\mathrm{q}\) (c) \(-\mathrm{p} \wedge \mathrm{q}\) (d) \(\mathrm{p} \wedge \mathrm{q}\)

If \(q\) is false and \(p \wedge q \leftrightarrow r\) is true, then which one of the following statements is a tautology? \(\quad\) (a) \((\mathrm{p} \vee \mathrm{r}) \rightarrow(\mathrm{p} \wedge \mathrm{r})\) (b) \((\mathrm{p} \wedge \mathrm{r}) \rightarrow(\mathrm{p} \vee \mathrm{r})\) (c) \(\mathrm{p} \wedge \mathrm{r}\) (d) \(\mathrm{p} \vee \mathrm{r}\)

Consider the following three statements: \(P\) : 5 is a prime number. \(Q: 7\) is a factor of 192 . R : L.C.M. of 5 and 7 is 35 . Then the truth value of which one of the following statements is true? \(\quad\) (a) \((\sim \mathrm{P}) \vee(\mathrm{Q} \wedge \mathrm{R})\) (b) \((P \wedge Q) \vee(\sim R)\) (c) \((\sim \mathrm{P}) \wedge(\sim \mathrm{Q} \wedge \mathrm{R})\) (d) \(\mathrm{P} \vee(\sim \mathrm{Q} \wedge \mathrm{R})\)

The logical statement \([\sim(\sim p \vee q) \vee(p \wedge r)] \wedge(\sim p \wedge r)\) is equivalent to: (a) \((\sim \mathrm{p} \wedge \sim \mathrm{q}) \wedge \mathrm{r}\) (b) \(\sim p \vee r\) (c) \((\mathrm{p} \wedge \mathrm{r}) \wedge \sim \mathrm{q}\) (d) \((\mathrm{p} \wedge \sim \mathrm{q}) \vee 1\)

The Boolean expression \(\sim(p \vee q) \vee(\sim p \wedge q)\) is equivalent to: \(\quad\) \(\begin{array}{llll}\text { (a) } \mathrm{p} & \text { (b) } \mathrm{q} & \text { (c) } \sim \mathrm{q} & \text { (d) } \sim \mathrm{p}\end{array}\)

If \(p \rightarrow(\sim p \vee \sim q)\) is false, then the truth values of \(p\) and \(q\) are respectively. (a) \(\mathrm{T}_{2} \mathrm{~F}\) (b) \(\mathrm{F}, \mathrm{F}\) (c) \(\mathrm{F}_{2} \mathrm{~T}\) (d) \(\mathrm{T}, \mathrm{T}\)

If \((\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r}) \rightarrow \sim p \vee q\) is false, then the truth values of \(\mathrm{p}, \mathrm{q}\) and \(\mathrm{r}\) are respectively \(\quad\) [Online April 15, 2018] (a) \(\mathrm{F}, \mathrm{T}, \mathrm{F}\) (b) \(\mathrm{T}, \mathrm{F}_{2} \mathrm{~T}\) \(\begin{array}{ll}\text { (c) } \mathrm{F}, \mathrm{F}, \mathrm{F} & \text { (d) } \mathrm{T}, \mathrm{T}, \mathrm{T}\end{array}\)

Which of the following is a tautology? (a) \((\sim \mathrm{p}) \wedge(\mathrm{p} \vee \mathrm{q}) \rightarrow \mathrm{q} \quad\) (b) \((\mathrm{q} \rightarrow \mathrm{p}) \vee \sim(\mathrm{p} \rightarrow \mathrm{q})\) (c) \((\sim q) \vee(p \wedge q) \rightarrow q\) (d) \((p \rightarrow q) \wedge(q \rightarrow p)\)

The following statement \((p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]\) is : \(\begin{array}{ll}\text { (a) a fallacy } & \text { (b) a tautology }\end{array}\) (c) equivalent to \(\sim \mathrm{p} \rightarrow \mathrm{q}\) (d) equivalent to \(\mathrm{p} \rightarrow \sim \mathrm{q}\)

The Boolean Expression \((\mathrm{p} \wedge-q) \vee q \vee(\sim p \wedge q)\) is equivalent to: (a) \(\mathrm{p} \vee \mathrm{q}\) (b) \(\mathrm{pv}-\mathrm{q}(\mathrm{c})-\mathrm{p} \wedge \mathrm{q} \quad\) (d) \(\mathrm{p} \wedge \mathrm{q}\)

The negation of \(\sim \mathrm{s} \vee(\sim \mathrm{r} \wedge \mathrm{s})\) is equivalent to: (a) \(s \vee(r \vee \sim s)\) (b) s\wedger (c) \(\mathrm{s} \wedge \sim \mathrm{r}\) (d) \(s \wedge(\mathrm{r} \wedge \sim \mathrm{s})\)

The statement \(-(p \leftrightarrow-q)\) is: (a) a tautology (b) a fallacy (c) eqivalent to \(p \leftrightarrow q\) (d) equivalent to \(\sim p \leftrightarrow q\)

Let \(p, q, r\) denote arbitrary statements. Then the logically equivalent of the statement \(p \Rightarrow(q \vee r)\) is: (a) \((p \vee q) \Rightarrow \mathrm{r}\) (b) \((p \Rightarrow q) \vee(p \Rightarrow r)\) (c) \((p \Rightarrow \sim q) \wedge(p \Rightarrow r)\) (d) \((p \Rightarrow q) \wedge(p \Rightarrow \sim r)\)

The proposition \(\sim(p \vee-q) \vee \sim(p \vee q)\) is logically equivalent to: (a) \(p\) (b) \(q\) (c) \(\sim p\) (d) \(\sim q\)

Consider Statement-1: \((p \wedge \sim q) \wedge(\sim p \wedge q)\) is a fallacy. Statement- \(2:(p \rightarrow q) \leftrightarrow(\sim q \rightarrow \sim p)\) is a tautology. (a) Statement- 1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1. (b) Statement- 1 is true; Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is false; Statement- 2 is true.

Let \(p\) and \(q\) be any two logical statements and \(\mathrm{r}: \mathrm{p} \rightarrow(-p \vee q) .\) If \(r\) has a truth value \(F\), then the truth values of \(p\) and \(q\) are respectively (a) \(\mathrm{F}_{2} \mathrm{~F}\) (b) \(\mathrm{T}_{2} \mathrm{~T}\) (c) \(\mathrm{T}_{2} \mathrm{~F}\) (d) \(\mathrm{F}, \mathrm{T}\)

For integers \(m\) and \(n\), both greater than 1 , consider the following three statements : \(P: m\) divides \(n\) \(Q: m\) divides \(n^{2}\) \(R: m\) isprime, then (a) \(Q \wedge R \rightarrow P\) (b) \(P \wedge Q \rightarrow R\) (c) \(Q \rightarrow R\) (d) \(Q \rightarrow P\)

The statement \(p \rightarrow(q \rightarrow p)\) is equivalent to : (a) \(p \rightarrow q\) (b) \(p \rightarrow(p \vee q)\) (c) \(p \rightarrow(p \rightarrow q)\) (d) \(p \rightarrow(p \wedge q)\)

Statement-1: The statement \(A \rightarrow(B \rightarrow A)\) is equivalent to \(A \rightarrow(A \vee B)\). Statement- \(2:\) The statement \(\sim[(\mathrm{A} \wedge \mathrm{B}) \rightarrow(\sim \mathrm{A} \vee \mathrm{B})]\) is a Tautology (a) Statement- 1 is false; Statement- 2 is true. (b) Statement- 1 is true; Statement- 2 is true; Statement- 2 is not correct explanation for Statement-1. (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is true; Statement- 2 is true; Statement- 2 is the correct explanation for Statement-1.

Let \(p\) and \(q\) be two Statements. Amongst the following, the Statement that is equivalent to \(p \rightarrow q\) is |Online May \(\mathbf{1 9 ,} \mathbf{2 0 1 2}\) ] \(\begin{array}{llll}\text { (a) } p \wedge \sim q & \text { (b) } \sim p \vee q & \text { (c) } \sim p \wedge q & \text { (d) } p \vee \sim q\end{array}\)

The logically equivalent preposition of \(p \Leftrightarrow q\) is (a) \((p \Rightarrow q) \wedge(q \Rightarrow p)\) (b) \(p \wedge q\) (c) \((p \wedge q) \vee(q \Rightarrow p)\) (d) \((p \wedge q) \Rightarrow(q \vee p)\)

The onlystatement among the following that is a tautology is \([ \mathrm{RS}]\) (a) \(A \wedge(A \vee B)\) (b) \(A \vee(A \wedge B)\) (c) \([\mathrm{A} \wedge(\mathrm{A} \rightarrow \mathrm{B})] \rightarrow \mathrm{B} \quad\) (d) \(\mathrm{B} \rightarrow[\mathrm{A} \wedge(\mathrm{A} \rightarrow \mathrm{B})]\)

Statement- \(1: \sim(p \leftrightarrow \sim q)\) is equivalent to \(p \leftrightarrow q\). Statement- \(2: \sim(p \leftrightarrow \sim q)\) is a tantology (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1. (b) Statement- 1 is true, Statement- 2 is false. (c) Statement- 1 is false, Statement- 2 is true. (d) Statement- 1 is true, Statement- 2 is true, Statement- 2 is a correct explanation for statement \(-1\)

The statement \(p \rightarrow(q \rightarrow p)\) is equivalent to (a) \(p \rightarrow(p \rightarrow q)\) (b) \(p \rightarrow(p \vee q)\) (c) \(p \rightarrow(p \wedge q)\) (d) \(p \rightarrow(p \leftrightarrow q)\)

Let \(p\) be the statement \(^{4} x\) is an irrational number", \(q\) be the statement "y is a transcendental number", and \(r\) be the statement " \(x\) is a rational number iff \(y\) is a transcendental number". \(\quad\) [2008] Statement-1: \(r\) is equivalent to either \(q\) or \(p\) Statement- \(2: r\) is equivalent to \(\sim(p \leftrightarrow \sim q)\). (a) Statement \(-1\) is false, Statement- 2 is true (b) Statement \(-1\) is true, Statement- 2 is true; Statement \(-2\) is a correct explanation for Statement- 1 (c) Statement \(-1\) is true, Statement- 2 is true; Statement \(-2\) is not a correct explanation for Statement-1 (d) Statement \(-1\) is true, Statement- 2 is false

Consider the statement: "For an integer \(\mathrm{n}\), if \(\mathrm{n}^{3}-1\) is even, then \(\mathrm{n}\) is odd." The contrapositive statement of this statement is: \(\quad\) (a) For an integer \(\mathrm{n}\), if \(\mathrm{n}\) is even, then \(\mathrm{n}^{3}-1\) is odd. (b) For an intetger \(\mathrm{n}\), if \(\mathrm{n}^{3}-1\) is not even, then \(\mathrm{n}\) is not odd. (c) For an integer \(\mathrm{n}\), if \(\mathrm{n}\) is even, then \(\mathrm{n}^{3}-1\) is even. (d) For an integer \(\mathrm{n}\), if \(\mathrm{n}\) is odd, then \(\mathrm{n}^{3}-1\) is even.

The statement \((p \rightarrow(q \rightarrow p)) \rightarrow(p \rightarrow(p \vee q))\) is: (a) equivalent to \((p \wedge q) \vee(\sim q)\) (b) a contradiction (c) equivalent to \((p \vee q) \wedge(\sim p)\) (d) a tautology

Contrapositive of the statement: 'If a function \(f\) is differentiable at \(a\), then it is also continuous at \(a^{\prime}\), is: (a) If a function \(f\) is continuous at \(a\), then it is not differentiable at \(a\). (b) If a function \(f\) is not continuous at \(a\), then it is not differentiable at \(a\). (c) If a function \(f\) is not continuous at \(a\), then it is differentiable at \(a\) (d) If a function \(f\) is continuous at \(a\), then it is differentiable at \(a\).

The contrapositive of the statement "If \(I\) reach the station in time, then \(I\) will catch the train" is: \(\quad\) (a) If \(I\) do not reach the station in time, then \(I\) will catch the train. (b) If \(I\) do not reach the station in time, then \(I\) will not catch the train. (c) If \(I\) will catch the train, then \(I\) reach the station in time. (d) If \(I\) will not catch the train, then \(I\) do not reach the station in time.

Negation of the statement: \(\sqrt{5}\) is an integer of 5 is irrational is: \(\quad\) (a) \(\sqrt{5}\) is not an integer or 5 is not irrational (b) \(\sqrt{5}\) is not an integer and 5 is not irrational (c) \(\sqrt{5}\) is irrational or 5 is an integer. (d) \(\sqrt{5}\) is an integer and 5 is irrational

The negation of the Boolean expression \(\sim s \vee(\sim r \wedge s)\) is equivalent to: \(\quad\) (a) \(\sim s \wedge \sim r\) (b) \(r\) (c) \(s \vee r\) (d) \(s \wedge r\)

For any two statements \(\mathrm{p}\) and \(\mathrm{q}\), the negation of the expression \(\mathrm{p} \vee(\sim \mathrm{p} \wedge \mathrm{q})\) is: (a) \(\sim \mathrm{p} \wedge \sim \mathrm{q}\) (b) \(\mathrm{p} \wedge q\) (c) \(\mathrm{p} \leftrightarrow q\) (d) \(\sim \mathrm{p} \vee \sim \mathrm{q}\)

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is : (a) If you are not a citizen of India, then you are not born in India. (b) If you are a citizen of India, then you are born in India. (c) If you are born in India, then you are not a citizen of India. (d) If you are not born in India, then you are not a citizen of India.

Contrapositive of the statement "If two numbers are not equal, then their squares are not equal", is : (a) If the squares of two numbers are not equal, then the numbers are equal. (b) If the squares of two numbers are equal, then the numbers are not equal. (c) If the squares of two numbers are equal, then the numbers are equal. (d) If the squares of two numbers are not equal, then the numbers are not equal.

. Consider the following two statements. Statement \(p:\) The value of \(\sin 120^{\circ}\) can be divided by taking \(\theta=240^{\circ}\) in the equation \(2 \sin \frac{\theta}{2}=\sqrt{1+\sin \theta}-\sqrt{1-\sin \theta}\). Statement \(q:\) Theangles \(A, B, C\) and \(D\) of any quadrilateral \(A B C D\) satisfy the equation \(\cos \left(\frac{1}{2}(A+C)\right)+\cos \left(\frac{1}{2}(B+D)\right)=0\) Then the truth values of \(p\) and \(q\) are respectively. (a) \(\mathrm{F}, \mathrm{T}\) (b) \(\mathrm{T}, \mathrm{T}\) (c) \(\mathrm{F}_{3} \mathrm{~F}\) (d) \(\mathrm{T}, \mathrm{F}\)