# Topic: Mathematical Reasoning (Test 2)

Topic: Mathematical Reasoning
Q.1
Let p and q be two propositions. Then the implication ∼(p↔q) is :
A. ∼p∨∼q
B. ∼p∧∼q
C. None of these
D. (p∧∼q)∨(∼p∧q)
Explaination / Solution:

∼(p↔q)≡(p∧∼q)∨(q∧∼p)

Workspace
Report
Q.2
Let p be the proposition : Mathematics is interesting and let q be the proposition that Mathematics is difficult, then the symbol p∧q means
A. Mathematics is interesting and Mathematics is difficult
B. Mathematics is interesting or Mathematics is difficult
C. Mathematics is interesting implies that Mathematics is difficult
D. Mathematics is interesting implies and is implied by Mathematics is difficult
Explaination / Solution:

using connective and for ∧

Workspace
Report
Q.3
Let p and q be two prepositions given by p : It is hot, q : He wants water Then , the verbal meaning of p→q is
A. If and only if it is hot, he wants water.
B. it is hot or he wants water.
C. If it is hot , then he wants water.
D. it is hot and he wants water.
Explaination / Solution:

→ symbol is replaced by if p then q

Workspace
Report
Q.4
Let p and q be two prepositions given by p : I have the raincoat, q : I can walk in rain. The compound proposition “ If I have the raincoat , then I can walk in the rain “ is represented by
A. p ∨q
B. p↔q
C. p→q
D. p ∨q
Explaination / Solution:

If then means an implication statement .hence we rewrite as p→q

Workspace
Report
Q.5
Which of the following is logically equivalent to (p∧q) ?
A. ∼(p∧∼q)
B. ∼(∼pv∼q)
C. p→∼q
D. (∼p∧∼q)
Explaination / Solution:

∼(∼(p∧q)) since∼p∨∼q≡∼(p∧q) ∼(∼p)≡p

Workspace
Report
Q.6
Which of the following is logically equivalent to p↔q ?
A. (p∧q)∧(q→p)
B. (p→q)∧(q→p)
C. (p→q)∨(q→p)
D. (p∧q)∧(q∨p)
Explaination / Solution:

By definition p↔q=(p→q)∧(q→p)

Workspace
Report
Q.7
(p∧∼q)∧(∼p∨q) is
A. both a tautology and a contradiction
B. a tautology
C. neither a tautology nor a contradiction
Explaination / Solution:

Since

F V F = F     Since

Hence contracdiction

Workspace
Report
Q.8
Negation of the statement q∨∼(p∧r) is
A. ∼q∧∼(p∧r)
B. ∼q∨(p∧r)
C. ∼q∧(p∧r)
D. ∼q→∼(p∧r)
Explaination / Solution:

∼(q∨∼(p∧r))   Since ∼(q∨r)≡∼q∧∼r
∼q∧(p∧r)

Workspace
Report
Q.9
The contrapositive of the statement “ if  then I get first class” is
A. If I do not get a first class , then
B. none of these
C. If I do not get a first class , then
D. If I get a first class , then
Explaination / Solution:

p:

q:I get first class

the contrapositive of . hence the answer is If I do not get a first class , then

Workspace
Report
Q.10
The inverse of the proposition (p∧∼q)→r is
A. (∼p∨q)→∼r
B. None of these
C. ∼r→∼p∨q
D. r→p∧∼q