Test: Mathematical Reasoning- 2

QUESTION: 1

The contrapositive of (p∨q)→ r is

A.
∼r→(p∧q)
B.
∼r→∼(p∨q)
C.
p→(p∧q)
D.
∼r→(∼p∧∼q)

Solution:

Contrapositive of p→q is ∼q→∼p.
∴ Contrapositive of (p∨q)⇒r is ∼r⇒∼(p∨q) i.e. ∼r⇒(∼p∧∼q).

QUESTION: 2

The contrapositive of p→(∼q→∼r) is

A.
(q∧∼r)→p
B.
(q∨∼r)∨p
C.
(∼q∧r)→∼p
D.
(q∧∼r)→∼p

Solution:

QUESTION: 3

Let p and q be two propositions. Then the inverse of the implication p→q is

A.
∼q→p
B.
∼p→∼q
C.
∼p→q
D.
p→∼q

Solution:

QUESTION: 4

Let p and q be two propositions. Then the contrapositive of the implication p→q is

A.
∼q→∼p
B.
∼p→∼q
C.
∼q→p
D.
p↔q

Solution:

QUESTION: 5

Let p and q be two propositions. Then the implication ∼(p↔q)∼(p↔q) is :

A.
(p∧∼q)∨(∼p∧q)
B.
∼p∨∼q
C.
∼p∧∼q
D.
none of these

Solution:

QUESTION: 6

p∧(q∧r) is logically equivalent to

A.
(p∨q)∧r
B.
(p∧q)∧r
C.
p→(q∧r)
D.
(p∨q)∨r

Solution:

QUESTION: 7

Which of the following proposition is a tautology ?

A.
∼p∧(∼p∨∼q)
B.
∼q∧(∼p∨∼q)
C.
(∼p∨∼q)∧(p∨∼q)
D.
(∼p∨∼q)∨(p∨∼q)

Solution:

QUESTION: 8

The negation of the compound statement p∨(∼p∨q) is

A.
(p∧q)∨p
B.
(p∧∼q)∧∼p
C.
(p∧∼q)∨p
D.
(p∧∼q)∨∼p

Solution:

QUESTION: 9

The proposition p→∼(p∧∼q) is

A.
a contradiction
B.
neither a contradiction nor a tautology
C.
a tautology
D.
none of these

Solution:

QUESTION: 10

The proposition (p→∼p)∧(∼p→p) is

A.
a contradiction
B.
a contradiction and a tautology
C.
neither a contradiction nor a tautology
D.
a tautology

Solution:

QUESTION: 11

Which of the following is equivalent to (p∧q) ?

A.
(p→q)
B.
∼(p→∼q)
C.
(p∧∼q)
D.
∼(p→q)

Solution:

QUESTION: 12

Which of the following is logically equivalent to ∼(∼p→q) ?

A.
(p∨∼q)
B.
(p∧q)
C.
(∼p∧∼q)
D.
(∼p∧q)

Solution:

∼(p⇒q)≡p∧∼q
∴ ∼(∼p⇒q) ≡∼p∧∼q

QUESTION: 13

The contrapositive of the inverse of p⇒ ~q is

A.
p ⇒ q
B.
~q ⇒ ~p
C.
~q ⇒ p
D.
~p ⇒ ~q

Solution:

The inverse of p ⇒ ∼q is ∼p ⇒ q
The contrapositive of ∼p ⇒ q is ∼q ⇒ p. [∴ Contrapositive of p ⇒ q is∼q ⇒ p.]

QUESTION: 14

p→q is logically equivalent to

A.
∼p→∼q
B.
p∧q
C.
p∧∼q
D.
∼q→∼p

Solution:

QUESTION: 15

The contrapositive of (∼p∧q)→ is

A.
(p∨q)→r
B.
r→(p∨∼q)
C.
(p∧q)→r
D.
none of these

Solution:

QUESTION: 16

The negation of p∧∼(q∧r) is

A.
∼p→(q∧r)
B.
p∨(q∧r)
C.
∼p∨(q∧r)
D.
∼p→(q∨∼r)

Solution:

QUESTION: 17

The negation of q∨∼(p∧r) is

A.
∼q∨(p∨r)
B.
∼q∧(p∧r)
C.
q∨(p∨r)
D.
∼q∧(p∨r)

Solution:

QUESTION: 18

Which of the following is a contradiction?

A.
(p→q)→p
B.
p∨(∼p∧q)
C.
(p∧q)∧(∼(p∨q))
D.
none of these

Solution:

QUESTION: 19

Which of the following is logically equivalent to (p∧q) ?

A.
∼(∼p∧∼q)
B.
∼(p→∼q)
C.
(p∧∼q)
D.
none of these

Solution:

QUESTION: 20

The negation of the proposition q∨∼(p∧r) is

A.
∼q∧(p∧r)
B.
q∧(p∧r)
C.
∼q∧(∼p∧∼r)
D.
none of these

Solution:

QUESTION: 21

If p→(q∨r) is false , then the truth values of p , q and r, are respectively

A.
T,F,F
B.
F,T,T
C.
F,F,F
D.
T,T,T

Solution:

QUESTION: 22

The statement p ⇒ p∨q is

A.
a contradiction
B.
both a tautology and contradiction
C.
neither a contradiction nor a tautology
D.
a tautology

Solution:

A tautology is a proposition is always true.

QUESTION: 23

The statement p∨qp∨q is

A.
a contradiction
B.
contingency
C.
a tautology
D.
none of these

Solution:

QUESTION: 24

Which of the following sentences is a statement ?

A.
2 is the smallest prime number
B.
Aarushi is a pretty girl
C.
What are you doing
D.
Oh ! It is amazing

Solution:

QUESTION: 25

Let p and q be two prepositions given by p : I take only bread and butter in breakfast. q : I do not take anything in breakfast. Then , the compound proposition “ I take only bread and butter in breakfast or I do not take anything “ is represented by