# Topic: Mathematical Reasoning (Test 5)

Topic: Mathematical Reasoning
Q.1
Which of the following is a proposition ?
A. Logic is an interesting subject
B. I am a lion
C. A triangle is a circle and 10 is a prime number
D. A half open door is half closed
Explaination / Solution:

it is a statement which is F.Hence it is a proposition.Other options are open sentences which are not propositions

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Q.2
Let p and q be two prepositions given by P : The sky is blue, q : Milk is white, Then , p and q is
A. The sky is white and milk is blue
B. if the sky is blue then , milk is white
C. The sky is blue or milk is white
D. The sky is blue and milk is white
Explaination / Solution:

using conjunction to combine statement

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Q.3
Let p and q be two prepositions given by p : To become an airforce officer one should be graduate. q : To become an airforce officer one should have good health. The compound proposition “ To become an airforce officer one should be a graduate and should have good health “ is represented by
A. p ∧q
B. p↔q
C. p ∨q
D. p→q
Explaination / Solution:

the statement before and is p .The statement after and is q.and is replaced by∧

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Q.4
∼(p∨q)∨(∼p∧q) is logically equivalent to
A. ∼p
B. p
C. q
D. ∼q
Explaination / Solution:

distributive law

since

since

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Q.5
∼p∨∼q is logically equivalent to
A. ∼p→∼q
B. p→∼q
C. p→∼q
D. p↔q
Explaination / Solution:

The answer is rule of negation for ∼(p→∼q)≡∼p∨∼q

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Q.6
The false statement in the following is
B. ∼(∼p)↔p is a tautology
C. p∨∼p is a tautology
Explaination / Solution:

p→q An implication statement,∼q→∼p a contrapositive statement. Implication and contrapositve have same meaning.hence they have identical values. p↔p≡T. hence the above statement will be true always.

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Q.7
Negation of the statement p→(q∧r) is
A. ∼p→(∼q∨r)
B. p→(∼q∨r)
C. ∼p∧(∼q∨∼r)
D. p∧(∼q∨∼r)
Explaination / Solution:

=     since

=

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Q.8
The contrapositive of (p∨q)→r is
A. p→(p∧q)
B. ∼r→(∼p∧∼q)
C. ∼r→(p∧q)
D. ∼r→∼(p∨q)
Explaination / Solution:

the contrapositive of p→q is∼q→∼p

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Q.9
Which of the following statement is a tautology ?
A. (∼q∧p)∧(p∧∼p)
B. (p∧q)∧(∼(p∧q))
C. (∼q∧p)∧q
D. (∼q∧p)∨(p∨∼p)
Explaination / Solution:

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Q.10
Let p and q be two propositions. Then the contrapositive of the implication p→q is
A. ∼q→∼p
B. ∼q→p
C. ∼p→∼q
D. p↔q