MCQ

MCQ 1012 Marks

Which of the following is always true?

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

Answer

Correct option: C.

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

(C)
$\begin{array}{l} p \rightarrow q \equiv \sim p \vee q \\ \therefore \quad \sim(p \rightarrow q) \equiv p \wedge \sim q\end{array}$

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MCQ 1022 Marks

The negation of $q \vee \sim(p \wedge \tau)$ is

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

Answer

Correct option: B.

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

(B)
$\begin{aligned} \sim[ q \vee \sim( p \wedge r )] & \equiv \sim q \wedge \sim(\sim( p \wedge r )) \\ & \equiv \sim q \wedge( p \wedge r )\end{aligned}$

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MCQ 1032 Marks

∼(p↔q) is equivalent to

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

Answer

Correct option: A.

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

(A)
We know that,
$p \leftrightarrow q \equiv(p \rightarrow q) \wedge(q \rightarrow p)$
$\therefore \quad \sim(p \leftrightarrow q) \equiv \sim[(p \rightarrow q) \wedge(q \rightarrow p)]$
$\equiv \sim(p \rightarrow q) \vee \sim(q \rightarrow p)…[By Demorgan's Law]$
$\begin{aligned} \equiv(p \wedge \sim q) & \vee(q \wedge \sim p) \\ & \quad \ldots[\because \sim(p \rightarrow q) \equiv p \wedge \sim q]\end{aligned}$

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MCQ 1042 Marks

Negation of the statement: '$\sqrt{5}$ is an integer or 5 is irrational' is

A
$\sqrt{5}$is not an integer or 5 is not irrational
B
$\sqrt{5}$is irrational or 5 is an integer
C
$\sqrt{5}$is an integer and 5 is irrational
✓
$\sqrt{5}$is not an integer and 5 is not irrational

Answer

Correct option: D.

$\sqrt{5}$is not an integer and 5 is not irrational

(D)
$p : \sqrt{5}$ is an integer
q : 5 is irrational
$\sim(p \vee q) \equiv \sim p \wedge \sim q$
∴ Option (D) is correct.

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MCQ 1052 Marks

The negation of the statement
"I like Mathematics and English" is

A
I do not like Mathematics and do not like English
B
I like Mathematics but do not like English
C
I do not like Mathematics but like English
✓
Either I do not like Mathematics or do not like English

Answer

Correct option: D.

Either I do not like Mathematics or do not like English

(D)
p : I like Mathematics
q : I like English.
$\sim(p \wedge q) \equiv \sim p \vee \sim q$
∴ Option (D) is correct.

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MCQ 1062 Marks

The negation of $(p \vee \sim q) \wedge q$ is

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

Answer

Correct option: C.

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

(C)
$\sim[( p \vee \sim q ) \wedge q ]$
$\equiv \sim(p \vee \sim q) \vee \sim q \quad \ldots .[$ De Morgan’s Law]
$\equiv(\sim p \wedge q ) \vee \sim q$

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MCQ 1072 Marks

Dual of the statement $(p \wedge q) \vee \neg q \equiv p \vee \neg q$ is

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

Answer

Correct option: C.

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

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MCQ 1082 Marks

The dual of the statement "Manoj has the job but he is not happy" is

✓
Manoj has the job or he is not happy.
B
Manoj has the job and he is not happy.
C
Manoj has the job and he is happy.
D
Manoj does not have the job and he is happy.

Answer

Correct option: A.

Manoj has the job or he is not happy.

(A)
p: Manoj has the job, q: he is not happy
The dual of $p \wedge q$ is $p \vee q$.
i.e., Manoj has the job or he is not happy.

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MCQ 1092 Marks

Using quantifiers $\forall, \exists$, convert the following open statement into true statement.
$' x+5-8, x \in N^{\prime}$

A
$\forall x \in N, x+5=8$
B
For every$x \in N, x+5>8$
✓
$\exists x \in N$, such that $x+5=8$
D
For every$x \in \mathrm{~N}, x+5<8$

Answer

Correct option: C.

$\exists x \in N$, such that $x+5=8$

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MCQ 1102 Marks

Using quantifier the open sentence '$x^2-4=32$' defined on W in converted into true statement as

A
$\forall x \in W, x^2-4=32$
B
$\exists x \in W$, such that $x^2-4 \leq 32$
C
$\forall x \in W, x^2-4>32$
✓
$\exists x \in W$, such that $x^2-4=32$

Answer

Correct option: D.

$\exists x \in W$, such that $x^2-4=32$

(D)
Option (D) is the required true statement, since $x=6 \in W$ satisfies $x^2-4=32$

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MCQ 1112 Marks

The proposition $p \rightarrow \sim(p\wedge \sim q)$ is a

A
contradiction
B
tautology
✓
contingency
D
none of these

Answer

Correct option: C.

contingency

(C)

p	q	∼q	$p \wedge \sim q$	$\sim( p \wedge \sim q )$	$p \rightarrow \sim( p \wedge \sim q )$
T	T	F	F	T	T
T	F	T	T	F	F
F	T	F	F	T	T
F	F	T	F	T	T

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MCQ 1122 Marks

The proposition $(p \rightarrow \sim p) \wedge(\sim p \rightarrow p)$ is a

A
Neither tautology nor contradiction
B
Tautology
C
Tautology and contradiction
✓
Contradiction

Answer

Correct option: D.

Contradiction

(D)

p	∼p	$p \rightarrow \sim p$	$\sim p \rightarrow p$	$(p \rightarrow \sim p) \wedge(\sim p \rightarrow p)$
T	F	F	T	F
F	T	T	F	F

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MCQ 1132 Marks

The proposition $(p \wedge q) \wedge(p \rightarrow\sim q$) is

✓
Contradiction
B
Tautology
C
Contingency
D
Tautology and Contradiction

Answer

Correct option: A.

Contradiction

(A)

p	q	∼q	$p \wedge q$	$p \rightarrow \sim q$	$( p \wedge q )\wedge( p \rightarrow \sim q )$
T	T	F	T	F	F
T	F	T	F	T	F
F	T	F	F	T	F
F	F	T	F	T	F

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MCQ 1142 Marks

The statement $(p \wedge q) \rightarrow p$ is

A
a contradiction
✓
a tautology
C
either (A) or (B)
D
a contingency

Answer

Correct option: B.

a tautology

(B)

p	q	$p \wedge q$	$(p \wedge q) \rightarrow p$
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	T

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MCQ 1152 Marks

When the compound statement is true for all its components then the statement is called

A
negation statement
✓
tautology statement.
C
contradiction statement.
D
contingency statement.

Answer

Correct option: B.

tautology statement.

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MCQ 1162 Marks

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

A
Tautology
✓
Contradiction
C
Tautology and contradiction
D
Contingency

Answer

Correct option: B.

Contradiction

(B) $(p \wedge \sim q) \wedge(\sim p \wedge q)$
$\equiv(p \wedge \sim p) \wedge(\sim q \wedge q) \ldots$...[Associative and commutative law]
$\equiv F \wedge F$...[Complement law]
$= F$
∴ Given statement is contradiction.

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MCQ 1172 Marks

The logically equivalent statement of $(p \wedge q) \vee(p \wedge r)$ is

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

Answer

Correct option: C.

$p \wedge(q \vee r)$

( C )
Distributive law

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MCQ 1182 Marks

The statement, 'If it is raining then I will go to college' is equivalent to

A
If it is not raining then I will not go to college.
✓
If I do not go to college, then it is not raining
C
If I go to college then it is raining.
D
Going to college depends on my mood

Answer

Correct option: B.

If I do not go to college, then it is not raining

(B)
$r$ : It is raining, $c$ : I will go to college.
The given statement is $r \rightarrow c \equiv \sim c \rightarrow \sim r$

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MCQ 1192 Marks

The logically equivalent statement of p↔q is

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

Answer

Correct option: C.

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

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MCQ 1202 Marks

Every conditional statement is equivalent to

✓
its contrapositive
B
its inverse
C
its converse
D
only itself

Answer

Correct option: A.

its contrapositive

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MCQ 1212 Marks

The statement p→(∼q) is equivalent to

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

Answer

Correct option: B.

$\sim q \vee \sim p$

(B)
$\begin{aligned} p \rightarrow(\sim q ) & =\sim p \vee \sim q \\ & \equiv \sim q \vee \sim p \end{aligned}$

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MCQ 1222 Marks

If p and q are two logical statements and A and B are two sets, then p →q corresponds to

✓
$A \subseteq B$
B
$A \cap B$
C
$A \cup B$
D
$\mathrm{A} \nsubseteq \mathrm{B}$

Answer

Correct option: A.

$A \subseteq B$

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MCQ 1232 Marks

The logical statement matrix ' $p \wedge q$ ' matrix can be related to the set theory's concept of

A
union of two sets
✓
intersection of two set
C
subset of a set
D
equality of two sets

Answer

Correct option: B.

intersection of two set

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MCQ 1242 Marks

Negation of a statement in logic corresponds to_________ in set theory.

A
empty set
B
null set
✓
complement of a set
D
universal set

Answer

Correct option: C.

complement of a set

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MCQ 1252 Marks

If p: Sita gets promotion,
q: Sita is transferred to Pune.
The verbal form of ∼p↔q is written as

A
Sita gets promotion and Sita gets transferred to Pune
B
Sita does not get promotion then Sita will be transferred to Pune.
C
Sita gets promotion if Sita is transferred to Pune.
✓
Sita does not get promotion if and only if Sita is transferred to Pune.

Answer

Correct option: D.

Sita does not get promotion if and only if Sita is transferred to Pune.

(D)
$\sim$ p: Sita does not get promotion and ' ↔' symbol indicates 'if and only if'.
i.e., Sita does not get promotion if and only if Sita is transferred to Pune.

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MCQ 1262 Marks

The contrapositive of the statement: "If a child concentrates then he learns" is

A
If a child does not concentrate he does not learn.
✓
If a child does not learn then he does not concentrate.
C
If a child practises then he learns
D
If a child concentrates, he does not forget

Answer

Correct option: B.

If a child does not learn then he does not concentrate.

(B)
Let p : A child concentrates
q : A child learns
Contrapositive of $p \rightarrow q$ is $\sim q \rightarrow \sim p$.
i.e., If a child does not learn then he does not concentrate

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MCQ 1272 Marks

The inverse of the statement "If you access the internet, then you have to pay the charges", is

✓
If you do not access the internet, then you do not have to pay the charges.
B
If you pay the charges, then you accessed the internet.
C
If you do not pay the charges, then you do not access the internet.
D
You have to pay the charges if and only if you access the internet

Answer

Correct option: A.

If you do not access the internet, then you do not have to pay the charges.

(A)
Let $p$ : You access the internet
q : You have to pay the charges
Inverse of $p \rightarrow q$ is $\sim p \rightarrow \sim q$
i.e., If you do not access the internet then you do not have to pay the charges.

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MCQ 1282 Marks

If Ram secures 100 marks in maths, then he will get a mobile. The converse is

A
If Ram gets a mobile, then he will not secure 100 marks in maths.
B
If Ram does not get a mobile, then he will secure 100 marks in maths.
✓
If Ram will get a mobile, then he secures 100 marks in maths.
D
None of these

Answer

Correct option: C.

If Ram will get a mobile, then he secures 100 marks in maths.

(C)
Let $p :$ Ram secures 100 marks in maths
q : Ram will get a mobile
Converse of $p \rightarrow q$ is $q \rightarrow p$
i.e., If Ram will get a mobile, then he secures 100 marks in maths.

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MCQ 1292 Marks

The converse of the statement "If x > y then x + a > y + a'' , is

A
If x < y then x + a < y + a
✓
If x + a > y + a then x > y
C
If x < y then x + a > y + a
D
If x > y then x + a < y + a

Answer

Correct option: B.

If x + a > y + a then x > y

(B)
Let p : x > y
q : x + a > y + a
Converse of $p \rightarrow q$ is $q \rightarrow p$
i.e., If $x+ a >y+ a$, then $x>y$

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MCQ 1302 Marks

The statement "If $x^2$ is not even then x is not even", is the converse of the statement

A
If$x^2$is odd, then x is even
✓
Ifx is not even, then $x^2$is not even
C
If x is even, then $x^2$is even
D
If x is odd, then $x^2$is even

Answer

Correct option: B.

Ifx is not even, then $x^2$is not even

(B)
Let $p : x^2$ is not even,
q : $x$ is not even
Converse of $p \rightarrow q$ is $q \rightarrow p$
i.e., If $x$ is not even then $x^2$ is not even

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MCQ 1312 Marks

If m: Rimi likes calculus.
n: Rimi opts for engineering branch.
Then the verbal form of m →n is

A
If Rimi opts for engineering branch then she likes calculus.
B
If Rimi likes calculus then she does not opt for engineering branch.
✓
If Rimi likes calculus then she opts for engineering branch
D
If Rimi likes engineering branch then she opts for calculus.

Answer

Correct option: C.

If Rimi likes calculus then she opts for engineering branch

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MCQ 1322 Marks

Assuming the first part of cach statement as p. second as q and the third as r, the statement 'if A, B,C are three distinet points, then either they are collinear or they form a triangle' in symbolic form is

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

Answer

Correct option: C.

$p \rightarrow(q \vee r)$

(C)
$p : A , B , C$, are distinct points
q : Points are collinear
r : Points form a triangle
$\therefore \quad p$ implies $( q$ or r $)$, i.e., $p \rightarrow( q \vee r )$

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MCQ 1332 Marks

A compound statement p →q is false only when

✓
p is true and q is false.
B
p is false but q is true.
C
atleast one of p or q is false.
D
both pand q are false.

Answer

Correct option: A.

p is true and q is false.

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MCQ 1342 Marks

If a: Vijay becomes a doctor,
b: Ajay is an engineer.
Then the statement 'Vijay becomes a doctor if and only if Ajay is an engineer' can be written in symbolic form as

A
$\mathrm{b} \leftrightarrow \sim \mathrm{a}$
✓
$a \leftrightarrow b$
C
$a \rightarrow b$
D
$\mathrm{b} \rightarrow \mathrm{a}$

Answer

Correct option: B.

$a \leftrightarrow b$

(B)
"if and only if" is expressed as ' ↔'
∴ symbolic form is $a \leftrightarrow b$.

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MCQ 1352 Marks

If d: driver is drunk, a: driver meets with an accident, translate the statement 'If the Driver is not drunk, then he cannot meet with an accident' into symbols

A
$\sim \mathrm{a} \rightarrow \sim \mathrm{d}$
✓
$\sim \mathrm{d} \rightarrow \sim \mathrm{a}$
C
$\sim d \wedge a$
D
$a \wedge \sim d$

Answer

Correct option: B.

$\sim \mathrm{d} \rightarrow \sim \mathrm{a}$

(B)
( $\sim d$ : Driver is not drunk) implies
( $\sim a$ : He cannot meet with an accident).
"Implies" is expressed as ' → '.
$\therefore$ symbolic form is $\sim d \rightarrow \sim$ a

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MCQ 1362 Marks

If p: Prakash passes the exam,
q: Papa will give him a bicycle.
Then the statement 'Prakash passing the exam, implies that his papa will give him a bicycle can be symbolically written as

✓
p→ q
B
$p \leftrightarrow q$
C
p ∧ q
D
p ∨ q

Answer

Correct option: A.

p→ q

(A)
"Implies" is expressed as ' → '.
∴ symbolic form is $p \rightarrow q$

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MCQ 1372 Marks

For the statements p and q 'p →q' is read as 'if p then q'. Here, the statement q is called

A
antecedent
✓
consequent
C
logical connective.
D
prime component.

Answer

Correct option: B.

consequent

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MCQ 1382 Marks

A compound statement p and 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 1392 Marks

A compound statement p or q is false only when

A
p is false.
B
q is false.
✓
both p and q are false.
D
depends on p and q.

Answer

Correct option: C.

both p and q are false.

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MCQ 1402 Marks

If p: Ram is lazy, q: Ram fails in the examination then the verbal form of ~p∨~q is

A
Ram is not lazy and he fails in the examination
✓
Ram is not lazy or he does not fail in the examination.
C
Ram is lazy or he does not fail in the examination.
D
Ram is not lazy and he does not fail in the examination.

Answer

Correct option: B.

Ram is not lazy or he does not fail in the examination.

(B) $\sim p$ : Ram is not lazy, $\sim q$ : Ram does not fail in the examination, 'v' indicates 'or'.

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MCQ 1412 Marks

Write verbally ∼p∨q where
p: She is beautiful; q: She is clever

A
She is beautiful but not clever
✓
She is not beautiful or she is clever
C
She is not beautiful or she is not clever
D
She is beautiful and clever.

Answer

Correct option: B.

She is not beautiful or she is clever

(B) $\sim p$ : She is not beautiful, ' v ' indicates 'or'.

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MCQ 1422 Marks

Let p be the proposition : Mathematics is interesting and let q be the proposition : Mathematics is difficult, then the symbol p^q means

A
Mathematics is interesting implies that Mathematics is difficult
B
Mathematics is interesting implies and is implied by Mathematics is difficult.
✓
Mathematics is interesting and Mathematics is difficult.
D
Mathematics is interesting or Mathematics is difficult.

Answer

Correct option: C.

Mathematics is interesting and Mathematics is difficult.

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MCQ 1432 Marks

Assuming the first part of the statement as p, second as q and the third as r, the statement 'Candidates are present, and voters are ready to vote but no ballot papers' in symbolic form is

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

Answer

Correct option: D.

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

(D)
$p :$ Candidates are present,
q : Voters are ready to vote
r : Ballot papers ⇒ $\sim$ r : no Ballot papers 'and' and 'but' are represented by ' $\wedge$ ' symbol.
$\therefore \quad(p \wedge q) \wedge \sim r$

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MCQ 1442 Marks

If p: Rohit is tall, q: Rohit is handsome, then the statement 'Rohit is tall or he is short and handsome' can be written symbolically as

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

Answer

Correct option: A.

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

(A)
p : Rohit is tall,
$\sim p$ : Rohit is short,
q : Rohit is handome,
'or' is expressed by ' $\vee$ ' symbol,
'and' is expressed by ' $\wedge$ ' symbol.
$ \therefore \quad p \vee(\sim p \wedge q)$

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MCQ 1452 Marks

If p: The sun has set, q: The moon has risen, then symbolically the statement 'The sun has not set or the moon has not risen' is written as