MCQ

MCQ 512 Marks

Given below are four statements along with their respective duals. Which dual statement is not correct?

A
$(p \vee q) \wedge(r \vee s),(p \wedge q) \vee(r \wedge s)$
B
$(p \vee \sim q) \wedge(\sim p),(p \wedge \sim q) \vee(\sim p)$
C
$(p \wedge q) \vee r,(p \vee q) \wedge r$
✓
$(p \vee q) \vee s,(p \wedge q) \vee s$

Answer

Correct option: D.

$(p \vee q) \vee s,(p \wedge q) \vee s$

(D)
Dual of ${ }^{\prime}(p \vee q) \vee{ }^{\prime}$ is ${ }^{\prime}(p \wedge q) \wedge s^{\prime}$.

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MCQ 522 Marks

The dual of $'(p \wedge t) \vee(c \wedge \sim q)^{\prime}$ where t is a V tautology and c is a contradiction, is

✓
$(p \vee c) \wedge(t \vee \sim q)$
B
$(\sim p \wedge c) \wedge(t \vee q)$
C
$(\sim p \vee c) \wedge(t \vee q)$
D
$(\sim p \vee t) \wedge(c \vee \sim q)$

Answer

Correct option: A.

$(p \vee c) \wedge(t \vee \sim q)$

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MCQ 532 Marks

Which of the following quantified statement is false?

A
$\exists x \in N$, such that $x+5 \leq 6$
B
$\forall x \in \mathrm{~N}, x^2 \not \leq 0$
✓
$\exists x \in \mathrm{~N}$, such that $x-1<0$
D
$\exists x \in \mathrm{~N}$, such that $x^2-3 x+2=0$

Answer

Correct option: C.

$\exists x \in \mathrm{~N}$, such that $x-1<0$

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MCQ 542 Marks

If A = {4, 5, 7, 9} determine which of the following quantified statement is true. A = \{4, 5, 7, 9\}

A
$\exists x \in A$, such that $x+4=7$
✓
$\forall x \in \mathrm{~A}, x+1 \leq 10$
C
$\forall x \in A, 2 x \leq 17$
D
$\exists x \in A$, such that $x+1>10$

Answer

Correct option: B.

$\forall x \in \mathrm{~A}, x+1 \leq 10$

(B)
Here, $x=4,5,7,9$ satisfies $x+1 \leq 10$
$\therefore \quad$ option (B) is correct.

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MCQ 552 Marks

Using quantifier the open sentence ' $x^2>0$ ' defined on N is converted into true statement as

✓
$\forall x \in \mathrm{~N}, x^2>0$
B
$\forall x \in N, x^2=0$
C
$\exists x \in N$, such that $x^2<0$
D
$\exists x \notin N$, such that $x^2<0$

Answer

Correct option: A.

$\forall x \in \mathrm{~N}, x^2>0$

(A)
Option (A) is the true statement, since square of every natural number is positive.

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MCQ 562 Marks

The false statement in the following is

A
$\mathrm{p} \wedge(\sim \mathrm{p})$ is a contradiction
B
$\mathrm{p} \vee(\sim \mathrm{p})$ is a tautology
C
$\sim(\sim p) \leftrightarrow p$ is tautology
✓
$(\mathrm{p} \rightarrow \mathrm{q}) \leftrightarrow(\sim \mathrm{q} \rightarrow \sim \mathrm{p})$ is a contradiction

Answer

Correct option: D.

$(\mathrm{p} \rightarrow \mathrm{q}) \leftrightarrow(\sim \mathrm{q} \rightarrow \sim \mathrm{p})$ is a contradiction

(D) $p \rightarrow q$ is logically equivalent to $\sim q \rightarrow \sim p$
$\therefore \quad(p \rightarrow q) \leftrightarrow(\sim q \rightarrow \sim p)$ is a tautology
...[Using Shortcut 8]
But, it is given contradiction.
Hence, it is false statement.

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MCQ 572 Marks

Which of the following statement is a contingency?

A
$(p \wedge \sim q) \vee \sim(p \wedge \sim q)$
B
$(p \wedge q) \leftrightarrow(\sim p \rightarrow \sim q)$
C
$(\sim q \wedge p) \vee(p \vee \sim p)$
D
$(q \rightarrow p) \vee(\sim p \leftrightarrow q)$

Answer

(D) Consider option (B)

p	q	$\sim p$	$\sim q$	$p \wedge q$	$\sim p \rightarrow \sim q$	$\begin{array}{l}(p \wedge q) \leftrightarrow \\ (\sim p \rightarrow \sim q)\end{array}$
T	T	F	F	T	T	T
T	F	F	T	F	T	F
F	T	T	F	F	F	T
F	F	T	T	F	T	F

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MCQ 582 Marks

Which of the following statement is contradiction?

A
$(p \wedge q) \rightarrow q$
✓
$(p \wedge \sim q) \wedge(p \rightarrow q)$
C
$p \rightarrow \sim(p \wedge \sim q)$
D
$(p \wedge q) \vee \sim q$

Answer

Correct option: B.

$(p \wedge \sim q) \wedge(p \rightarrow q)$

(B)
consider option (B)
$(p \wedge \sim q) \wedge(p \rightarrow q)$
$\equiv \sim(p \rightarrow q) \wedge(p \rightarrow q)$
$\equiv F \quad \ldots[$ Complement Law $]$

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MCQ 592 Marks

$(\sim p \wedge \sim q) \wedge(q \wedge r)$ is a

A
tautology
B
contingency
✓
contradiction
D
neither tautology nor contradiction

Answer

Correct option: C.

contradiction

(C)
$(\sim p \wedge \sim q) \wedge(q \wedge r)$
$\equiv \sim p \wedge(\sim q \wedge q) \wedge r \quad \ldots[$ Associative Law $]$
$\equiv \sim p \wedge F \wedge r \quad \ldots[$ Complement Law $]$
$\equiv F \quad \ldots[$ Indentity Law $]$
∴ Given statement is contradiction.

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MCQ 602 Marks

Which of the following is a tautology?

A
$p \rightarrow(p \wedge q)$
B
$q \wedge(p \rightarrow q)$
✓
$\sim(p \rightarrow q) \leftrightarrow p \wedge \sim q$
D
$(p \wedge q) \leftrightarrow \sim q$

Answer

Correct option: C.

$\sim(p \rightarrow q) \leftrightarrow p \wedge \sim q$

(C)
Consider option (C)
$\sim(p \rightarrow q) \leftrightarrow(p \wedge \sim q)$
$\equiv( p \wedge \sim q ) \leftrightarrow( p \wedge \sim q )$
$\ldots[\because \sim(p \rightarrow q) \equiv p \wedge \sim q]$
$\equiv T$
...[Using Shortcut 8]

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MCQ 612 Marks

Which of the following statements is a tautology?

A
$\sim(p \wedge \sim q) \rightarrow(p \vee q)$
B
$(\sim p \vee \sim q) \rightarrow(p \wedge q)$
C
$p \vee(\sim q) \rightarrow(p \wedge q)$
✓
$\sim(p \vee \sim q) \rightarrow(p \vee q)$

Answer

Correct option: D.

$\sim(p \vee \sim q) \rightarrow(p \vee q)$

(D)Consider option (D)
$
\sim(p \vee \sim q) \rightarrow(p \vee q)
$
$\equiv(p \vee \sim q) \vee(p \vee q) \quad \ldots[\because p \rightarrow q \equiv \sim p \vee q]$
$\equiv p \vee(\sim q \vee q ) \quad \ldots[$ Absorption law $]$
$\equiv p \vee T \quad \ldots[$ Complement law $]$
$\equiv T$...[Identity law]

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MCQ 622 Marks

Which one of the following statements is not a tautology?

A
$p \rightarrow(p \vee q)$
B
$(p \wedge q) \rightarrow(\sim p \vee q)$
C
$(p \wedge q) \rightarrow p$
✓
$(p \vee q) \rightarrow(p \vee \sim q)$

Answer

Correct option: D.

$(p \vee q) \rightarrow(p \vee \sim q)$

(D)
Consider option (D)

p	q	$p \vee q$	$\sim q$	$p \vee \sim q$	$\begin{array}{l}(p \vee q) \rightarrow \\ (p \vee \sim q)\end{array}$
T	T	T	F	T	T
T	F	T	T	T	T
F	T	T	F	F	F
F	F	F	T	T	T

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MCQ 632 Marks

$\sim(\sim p) \leftrightarrow p$ is

✓
a tautology
B
a contradiction
C
neither a contradiction nor a tautology
D
none of these

Answer

Correct option: A.

a tautology

(A)
$\sim(\sim p) \rightarrow p \equiv p \rightarrow p \quad \ldots[\because \sim(\sim p) \equiv p]$
∴ Using Shortcut 1 , we get
$p \rightarrow p \equiv T$
$\therefore \quad \sim(\sim p) \leftrightarrow p$ is a tautology.

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MCQ 642 Marks

Which of the following is not true for any two statements p and q?

A
$\sim[p \vee(\sim q)] \equiv \sim p \wedge q$
B
$(n \vee q) \vee(\sim q)$ is a tautology
C
$\neg(\mathrm{p} \wedge \sim \mathrm{p})$ is a tautology
✓
$\sim(p \vee q) \equiv \sim p \vee \sim q$

Answer

Correct option: D.

$\sim(p \vee q) \equiv \sim p \vee \sim q$

(D)
$\sim(p \vee q) \equiv \sim p \vee \sim q$ is not true as it contradicts De Morgan's law.
∴ Option (D) is not true.

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MCQ 652 Marks

The logical statement $(p \rightarrow q) \wedge(q \rightarrow \sim p)$ is equivalent to:

A
p
B
∼q.
C
q
✓
∼p

Answer

Correct option: D.

∼p

(D)
$(p \rightarrow q) \wedge(q \rightarrow \sim p)$

p	q	$\sim p$	$p \rightarrow q$	$q \rightarrow \sim p$	$\begin{array}{c}(p \rightarrow q) \\ \wedge \\ (q \rightarrow \sim p)\end{array}$
T	T	F	T	F	F
T	F	F	F	T	F
F	T	T	T	T	T
F	F	T	T	T	T

$\therefore \quad(p \rightarrow q) \wedge(p \rightarrow \sim q)=\sim p$

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MCQ 662 Marks

The statement pattern $(p \wedge q) \wedge[\sim r \vee(p \wedge q)]$ $\vee(\sim p \wedge q)$ is equivalent to

A
$p \wedge q$
B
r
C
p
✓
q

Answer

Correct option: D.

(D)
$(p \wedge q) \wedge[\sim r \vee(p \wedge q)] \vee(\sim p \wedge q)$
$\equiv( p \wedge q ) \vee(\sim p \wedge q ) \quad \ldots[$ Absorption law $]$
$\equiv( p \vee \sim p ) \wedge q \quad \ldots[$ Distributive law $]$
$\equiv T \wedge q \quad \ldots[$ Complement law $]$
$\equiv q \quad \ldots[$ Identity law $]$

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MCQ 672 Marks

Which of the following is NOT equivalent to p→q.

A
p is sufficient for q
B
ponly if q
C
q is necessary for p
✓
q only if p

Answer

Correct option: D.

q only if p

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MCQ 682 Marks

Which of the following is true?

A
$p \wedge \sim p \equiv T$
B
$p \vee \sim p \equiv F$
C
$p \rightarrow q \equiv q \rightarrow p$
✓
$p \rightarrow q \equiv(\sim q) \rightarrow(\sim p)$

Answer

Correct option: D.

$p \rightarrow q \equiv(\sim q) \rightarrow(\sim p)$

(D)
$(\sim q) \rightarrow(\sim p)$ is contrapositive of $p \rightarrow q$ and hence both are logically equivalent to each other.

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MCQ 692 Marks

$p \wedge(p \rightarrow q)$ is logically equivalent to

A
$p \vee q$
B
$\sim p \vee q$
✓
$p \wedge q$
D
$\mathrm{p} \vee \sim \mathrm{q}$

Answer

Correct option: C.

$p \wedge q$

(C)
$\begin{array}{l}p \wedge(p \rightarrow q) \\ \equiv p \wedge(\sim p \vee q)\end{array}$
$\equiv( p \wedge \sim p ) \vee( p \wedge q ) \quad \ldots[$ Distributive law $]$
$\equiv F \vee(p \wedge q) \quad \ldots$ [Complement law]
$
\equiv p \wedge q\quad \ldots$ [Identity law]

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MCQ 702 Marks

Find which of the following statements convey the same meanings?
i. If it is the bride's dress then it has to be red.
ii. If it is not bride's dress then it cannot be red.
iii. If it is a red dress then it must be the bride's dress.
iv. If it is not a red dress then it can't be the bride's dress.

✓
(i, iv) and (ii, iii)
B
(i, ii) and (iii, iv)
C
(i), (ii), (iii)
D
(i, iii) and (ii, iv)

Answer

Correct option: A.

(i, iv) and (ii, iii)

(A)
b: It is the bride's dress
r : It is the red dress
i. $b \rightarrow r$
ii. $\sim b \rightarrow \sim r$
iii. $r \rightarrow b$
iv. $\sim r \rightarrow \sim b$
(i) and (iv) have the same meaning,
(ii) and (iii) have the same meaning.

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MCQ 712 Marks

Find out which of the following statements have the same meaning:
i. If Seema solves a problem then she is happy.
ii. If Seema does not solve a problem then she is not happy.
iii. If Seema is not happy then she hasn't solved the problem.
iv. If Seema is happy then she has solved the problem.

A
(i, ii) and (iii, iv)
B
i, ii, iii
✓
(i, iii) and (ii, iv)
D
ii, iii, iv

Answer

Correct option: C.

(i, iii) and (ii, iv)

(C)
p: Seema solves a problem
q : She is happy
i. $p \rightarrow q$
ii. $\sim p \rightarrow \sim q$
iii. $\sim q \rightarrow \sim p$
iv. $q \rightarrow p$
(i) and (iii) have the same meaning,
(ii) and (iv) have the same meaning.

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MCQ 722 Marks

The statement $p \rightarrow(q \rightarrow p)$ is equivalent to

A
$p \rightarrow(p \wedge q)$
B
$p \rightarrow(p \leftrightarrow q)$
C
$p \rightarrow(p \rightarrow q)$
D
$p \rightarrow(p \vee q)$

Answer

Consider option (D)

l	2	3	4	5	6
p	q	$q \rightarrow p$	$\begin{array}{c}p \rightarrow \\ (q \rightarrow p)\end{array}$	$p \vee q$	$\begin{array}{c}p \rightarrow \\ (p \vee q)\end{array}$
T	T	T	T	T	T
T	F	T	T	T	T
F	T	F	T	T	T
F	F	T	T	F	T

The entries in the columns 4 and 6 are identical.
$\therefore p \rightarrow(q \rightarrow p)=p \rightarrow(p \vee q)$

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MCQ 732 Marks

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is

A
If you are a citizen of India, then you are born in India.
B
If you are born in India, then you are not a citizen of India.
✓
If you are not a citizen of India, then you are not born in India.
D
If you are not born in India, then you are not a citizen of India.

Answer

Correct option: C.

If you are not a citizen of India, then you are not born in India.

(C)
Contrapositive of $p \rightarrow q$ is $\sim q \rightarrow \sim p$
The contrapositive of the given statement is "If you are not a citizen of India, then you are not born in India".

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MCQ 742 Marks

The converse of 'If x is zero then we cannot divide by x' is

✓
If we cannot divide by x then x is zero.
B
If we divide by x then x is non-zero.
C
If x is non-zero then we can divide by x.
D
If we cannot divide by x then x is non-zero.

Answer

Correct option: A.

If we cannot divide by x then x is zero.

(A)
Let $p : x$ is zero
q : we cannot divide by $x$
Converse of $p \rightarrow q$ is $q \rightarrow p$.
∴ Converse of the given statement is 'If we cannot divide by $x$ then $x$ is zero'.

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MCQ 752 Marks

The converse of the contrapositive of $\mathrm{p} \rightarrow \mathrm{q}$ is

A
$\sim p \rightarrow q$
B
$p \rightarrow \sim q$
✓
$\sim p \rightarrow \sim q$
D
$\sim q \rightarrow p$

Answer

Correct option: C.

$\sim p \rightarrow \sim q$

(C)
Given $p \rightarrow q$
Its contrapositive is $\sim q \rightarrow \sim p$ and converse of the contrapositive is $\sim p \rightarrow \sim q$.

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MCQ 762 Marks

If $\sim q \vee p$ is F, then which of the following is correct?

A
$p \leftrightarrow q$ is $T$
✓
$p \rightarrow q$ is $T$
C
$q \rightarrow p$ is $T$
D
$p \rightarrow q$ is $F$

Answer

Correct option: B.

$p \rightarrow q$ is $T$

(B)
The truth value of statement $\sim q \vee p$ is $F$
$\therefore \quad \sim q \equiv F$ and $p \equiv F$
$\begin{array}{l}\therefore \quad p \equiv F \text { and } q \equiv T \\ \therefore \quad p \rightarrow q \text { is } T\end{array}$
Alternate Method:

p	q	$ \sim q $	$\sim q \vee p$	$p \leftrightarrow q$	$p \rightarrow q$	$q \rightarrow p$
T	T	F	T	T	T	T
T	F	T	T	F	F	T
F	T	F	F	F	T	F
F	F	T	T	T	T	T

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MCQ 772 Marks

If $p \rightarrow(p \wedge \sim q)$ is false, the truth values of p and q are respectively

A
F, F
B
T, F
✓
T, T
D
F, T

Answer

Correct option: C.

T, T

(C)
The truth value of statement $p \rightarrow(p \wedge \sim q)$ is $F$
$\therefore \quad p \equiv T$ and $(p \wedge \sim q) \equiv F$
$\therefore \quad p = T$ and $q = T$

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MCQ 782 Marks

If $p \rightarrow(\sim p \vee q)$ is false, the truth values of p and q are respectively

A
F, T
B
F. F
C
T, T
✓
T, F

Answer

Correct option: D.

T, F

(D)

p	q	$\sim p$	$\sim p \vee q$	$p \rightarrow(\sim p \vee q)$
T	T	F	T	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

$\therefore$ From the table $p \rightarrow(\sim p \vee q)$ is false when $p$ is true and $q$ is false.

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MCQ 792 Marks

If p, q are true and r is false statement then which of the following is true statement?

A
$(p \wedge q) \vee r$ is $F$
B
$(p \wedge q) \rightarrow r$ is $T$
✓
$(p \vee q) \wedge(p \vee r)$ is $T$
D
$(p \rightarrow q) \leftrightarrow(p \rightarrow r)$ is $T$

Answer

Correct option: C.

$(p \vee q) \wedge(p \vee r)$ is $T$

(C)$
\begin{aligned}
(p \vee q) \wedge(p \vee r) & \equiv(T \vee T) \wedge(T \vee F) \\
& \equiv T \wedge T \equiv T
\end{aligned}
$
∴ option (C) is correct.

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MCQ 802 Marks

Given that p is ‘false’ and q is 'true' then the statement which is 'false' is

✓
$\sim \mathrm{p} \rightarrow \sim \mathrm{q}$
B
$p \rightarrow(q \wedge p)$
C
$p \rightarrow \sim q$
D
$q \rightarrow \sim p$

Answer

Correct option: A.

$\sim \mathrm{p} \rightarrow \sim \mathrm{q}$

(A)
$\begin{array}{l}\sim p \rightarrow \sim q \equiv \sim F \rightarrow \sim T \equiv T \rightarrow F \equiv F \\ p \rightarrow(q \wedge p) \equiv F \rightarrow(T \wedge F) \equiv F \rightarrow F \equiv T \\ p \rightarrow \sim q \equiv F \rightarrow \sim T \equiv F \rightarrow F \equiv T \\ q \rightarrow \sim p \equiv T \rightarrow \sim F \equiv T \rightarrow T \equiv T\end{array}$

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MCQ 812 Marks

If p is false and q is true, then

A
$\mathrm{p} \wedge \mathrm{q}$ is true
B
$\mathrm{p} \vee \sim \mathrm{q}$ is true
C
$\mathrm{q} \rightarrow \mathrm{p}$ is true
✓
$p \rightarrow q$ is true

Answer

Correct option: D.

$p \rightarrow q$ is true

(D)
$\begin{array}{l} p \wedge q = F \wedge T = F \\ p \vee \sim q \equiv F \vee \sim T \equiv F \vee F \equiv F \\ q \rightarrow p \equiv T \rightarrow F \equiv F \\ p \rightarrow q \equiv F \rightarrow T \equiv T \end{array}$

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MCQ 822 Marks

If p is true and q is false then the truth values of $(p \rightarrow q) \leftrightarrow(\sim q \rightarrow \sim p)$ and $(\sim p \vee q) /(\sim q / 2)$ are respectively

A
F, F
B
F, T
✓
T, F
D
T, T

Answer

Correct option: C.

T, F

(C)
$\begin{array}{l}( p \rightarrow q ) \leftrightarrow(\sim q \rightarrow \sim p ) \\ =( T \rightarrow F ) \leftrightarrow \\ \quad(\sim F \rightarrow \sim T ) \\ \equiv F \leftrightarrow( T \rightarrow F ) \\ \equiv F \leftrightarrow F \\ \equiv F \end{array}$
$\begin{array}{l}(\sim p \vee q) \wedge(\sim q \vee p) \\ \equiv(\sim T \vee F) \wedge(\sim F \vee T) \\ \equiv(F \vee F) \wedge(T \vee T) \\ \equiv F \wedge T \\ \equiv F\end{array}$

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MCQ 832 Marks

If p and q have truth value 'F', then the truth values of (∼p ∨ q)↔ ∼(p ∧ q) and ∼p↔(p→∼q) are respectively

✓
T, T
B
F, F
C
T, F
D
F, T

Answer

Correct option: A.

T, T

(A)
$\begin{array}{l}(\sim p \vee q) \leftrightarrow \sim(p \wedge q) \\ \equiv(\sim F \vee F) \leftrightarrow \sim(F \wedge F) \\ \equiv(T \vee F) \leftrightarrow \sim F \\ \equiv T \leftrightarrow T \\ \equiv T\end{array}$
$\begin{array}{l}\sim p \leftrightarrow( p \rightarrow \sim q ) \\ \equiv \sim F \leftrightarrow( F \rightarrow \sim F ) \\ \equiv T \leftrightarrow( F \rightarrow T ) \\ \equiv T \leftrightarrow T \\ \equiv T \end{array}$

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MCQ 842 Marks

Given 'p'and 'q' as true and 'r' as false, the truth values of $\sim p \wedge(q \vee \sim r)$ and $(p \rightarrow q) \wedge r$ are respectively

A
T, F
✓
F,F
C
T,T
D
F, T

Answer

Correct option: B.

F,F

(B)
$\sim p \wedge(q \vee \sim r)$
$\equiv \sim T \wedge( T \vee \sim F )$
$\equiv F \wedge( T \vee T )$
$\begin{array}{l}\equiv F \wedge T \\ \equiv F \end{array}$
$\begin{array}{l}(p \rightarrow q) \wedge r \\ \equiv(T \rightarrow T) \wedge F \\ \equiv T \wedge F \\ \equiv F\end{array}$

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MCQ 852 Marks

Let p: roses are red and q : the sun is a star. Then the verbal translation of $(\sim p) \vee q$ is

A
Roses are not red and the sun is not a star.
B
It is not true that roses are red or the sun is not a star.
C
It is not true that roses are red and the sun is not a star.
✓
Roses are not red or the sun is a star.

Answer

Correct option: D.

Roses are not red or the sun is a star.

(D)
$\sim p$ : roses are not red
$\sim p \vee q:$ roses are not red or the sun is a star.

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MCQ 862 Marks

The symbolic form of the statement 'It is not true that intelligent persons are neither polite nor helpful' is

A
$\sim(p \vee q)$
✓
$\sim(\sim p \wedge \sim q)$
C
$\sim(\sim p \vee \sim q)$
D
$\sim(p \wedge q)$

Answer

Correct option: B.

$\sim(\sim p \wedge \sim q)$

(B)
p : Intelligent persons are polite.
q : Intelligent persons are helpful.
∴ Symbolic form: $\sim(\sim p \wedge \sim q)$

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MCQ 872 Marks

If first part of the sentence is p and the second is q, then the symbolic form of the statement 'It is not true that Physics is not interesting or difficult' is

A
$\sim(\sim p \wedge q)$
B
$(\sim p \vee q)$
C
$(\sim p \vee \sim q)$
✓
$\sim(\sim p \vee q)$

Answer

Correct option: D.

$\sim(\sim p \vee q)$

(D)
p : Physics is interesting.
q : Physics is difficult.
$\therefore \quad$ Symbolic form: $\sim(\sim p \vee q)$

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MCQ 882 Marks

Assuming the first part of the sentence as p and the second as q, write the following statement symbolically:
'Irrespective of one being lucky or not, one should not stop working'

A
$(p \wedge \sim p) \vee q$
B
$(p \vee \sim p) \wedge q$
C
$(p \vee \sim p) \wedge \sim q$
D
$(p \wedge \sim p) \vee \sim q$

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MCQ 892 Marks

Which of the following is an incorrect statement in logic?

A
Multiply the numbers 3 and 10.
✓
3 times 10 is equal to 40.
C
What is the product of 3 and 10?
D
10 times 3 is equal to 30.

Answer

Correct option: B.

3 times 10 is equal to 40.

(B)
'Incorrect statement' means a statement in logic with truth value false.
Options (A) and (C) are not statements in logic.
Option (D) has truth value True.
Option (B) is a statement in logic with truth value false.

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MCQ 902 Marks

Which of the following is not a correct statement?

A
Mathematics is interesting.
✓
$\sqrt{3}$ is a prime.
C
$\sqrt{2}$ is irrational.
D
The sun is a star.

Answer

Correct option: B.

$\sqrt{3}$ is a prime.

(B)
"Not a correct statement" means it is a statement whose truth value is false.
Option (A) is not a statement.
Options (C) and (D) are statements with truth value true.
' $\sqrt{3}$ is a prime' is false statement.
∴ Option (B) is correct.

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MCQ 912 Marks

The negation of 'If it is sunday then it is a holiday' is

A
It is a holiday but not a sunday.
B
No sunday then no holiday.
✓
It is sunday, but it is not a holiday.
D
No holiday therefore no sunday.

Answer

Correct option: C.

It is sunday, but it is not a holiday.

(C) p : It is Sunday
q : It is a holiday
Symbolic from p → q
$\sim( p \rightarrow q ) \equiv p \wedge \sim q$
i.e. , it is sunday, but is not a holiday

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MCQ 922 Marks

Which of the following statements is a tautology?

A
$(\sim q \wedge p) \wedge q$
B
$(\sim q \wedge p) \wedge q$
✓
$(\sim q \wedge p) \vee(p \vee \sim p)$
D
$(p \wedge q) \wedge(\sim(p \wedge q))$

Answer

Correct option: C.

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

(C)
Since $p \vee \sim p \equiv T$,
$(\sim q \wedge p) \vee(p \vee \sim p) \equiv(\sim q \wedge p) \vee T$
$\equiv T$
∴ $(\sim q \wedge p) \vee(p \vee \sim p)$ is a tautology.

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MCQ 932 Marks

The Boolean Expression $(p \wedge \sim q) \vee q \vee(\sim p \wedge q)$ is equivalent to :

A
p ^ q
✓
p v q
C
$p \vee \sim q$
D
$\sim p \wedge q$

Answer

Correct option: B.

p v q

(B)
$(p \wedge \sim q) \vee q \vee(\sim p \wedge q)$
$\equiv[(p \vee q) \wedge(\sim q \vee q)] \vee(\sim p \wedge q)$...[Distributive law]
$\equiv[(p \vee q) \wedge T] \vee(\sim p \wedge q)$...[Complement law]
$\equiv(p \vee q) \vee(\sim p \wedge q)$ ...[Identity law]
$\equiv(p \vee q \vee \sim p) \wedge(p \vee q \vee q)$...[Distributive law]
$\equiv(T \vee q) \wedge(p \vee q)$...[Complement law and Idempotent law]
$\equiv T \wedge( p \vee q )$...[Identity law]
$\equiv p \vee q$...[Identity law]

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MCQ 942 Marks

A compound statement p or q is true only when

A
p is true.
B
q is true.
✓
both p and q are true.
D
none of p and q is true.

Answer

Correct option: C.

both p and q are true.

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MCQ 952 Marks

The switching circuit

✓
$(p \wedge q) \vee(\sim p) \vee(p \wedge \sim q)$
B
$(p \vee q) \vee(\sim p) \vee(p \wedge \sim q)$
C
$(p \wedge q) \wedge(\sim p) \vee(p \wedge \sim q)$
D
$(p \vee q) \wedge(\sim p) \vee(p \wedge \sim q)$

Answer

Correct option: A.

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

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MCQ 962 Marks

If the current flows through the given circuit, then it is expressed symbolically as,

✓
$(p \wedge q) \vee r$
B
$(p \wedge q)$
C
$(p \vee q)$
D
$(p \vee q) \wedge r$

Answer

Correct option: A.

$(p \wedge q) \vee r$

(A)
Current will flow in the circuit if switch $p$ and $q$ are closed or switch $r$ is closed.
It is represented by $( p \wedge q ) \vee r$.
∴ option (A) is correct.

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MCQ 972 Marks

The switching circuit for the statement $p \wedge q \wedge T$ is

Answer

Correct option: A.

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MCQ 982 Marks

When does the current flow through the following circuit.

A
p. q should be closed and r is open
B
р, q,r should be open
✓
p, q, rshould be closed
D
none of these

Answer

Correct option: C.

p, q, rshould be closed

(C)
The current will flow through the circuit if $p , q , r$ are closed or $p , q ^{\prime}, r$ are closed.
$\therefore \quad$ option (C) is the correct answer.

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MCQ 992 Marks

The negation of the statement "All continuous functions are differentiable"

A
Some continuous functions are differentiable
B
All differentiable functions are continuous
C
All continuous functions are not differentiable
✓
Some continuous functions are not differentiable

Answer

Correct option: D.

Some continuous functions are not differentiable

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MCQ 1002 Marks

The negation of 'For every natural number x, $x+5>4$' is.

A
$\forall x \in \mathrm{~N}, x+5<4$
B
$\forall x \in \mathrm{~N}, x-5<4$
C
For every integer $x, x+5<4$
✓
There exists a natural number x, for which $x+5 \leq 4$

Answer

Correct option: D.

There exists a natural number x, for which $x+5 \leq 4$

(D)
Given statement is ' $\forall x \in N, x+5>4$ '
$\begin{aligned}
\therefore & \sim[\forall x \in N, x+5>4]
\\ & \equiv \exists x \in N, \text { such that } x+5 \leq 4\end{aligned}$
i.e., there exists a natural number $x$, for which $x+5 \leq 4$

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Maths MCQ