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)
Answer : Option D
Explaination / Solution:

∼(p↔q)≡(p∧∼q)∨(q∧∼p)

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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
Answer : Option A
Explaination / Solution:

using connective and for ∧

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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.
Answer : Option C
Explaination / Solution:

→ symbol is replaced by if p then q

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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
Answer : Option C
Explaination / Solution:

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

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

∼(∼(p∧q)) since∼p∨∼q≡∼(p∧q) ∼(∼p)≡p

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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)
Answer : Option B
Explaination / Solution:

By definition p↔q=(p→q)∧(q→p)

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

     Since 

F V F = F     Since       

Hence contracdiction

 


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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)
Answer : Option C
Explaination / Solution:

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

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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 
Answer : Option C
Explaination / Solution:

p:

q:I get first class

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


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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
Answer : Option A
Explaination / Solution:

         

   Since 


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