# Topic: Mathematical Reasoning (Test 1)

Topic: Mathematical Reasoning
Q.1
For any three propositions p , q , and r , the proposition (p∧q)∧(q∧r) is true , when
A. p, q are true and r is false
B. p, q , r are all false
C. p,q , r are all true
D. p is true and q, r are false
Answer : Option C
Explaination / Solution:

hence p=T , q=T, r=T

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Q.2
Which of the following proposition is a tautology ?
A. (∼p∨∼q)∧(p∨∼q)
B. (∼p∨∼q)∨(p∨∼q)
C. ∼p∧(∼p∨∼q)
D. ∼q∧(∼p∨∼q)
Answer : Option B
Explaination / Solution:

and          Associative law

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Q.3
Let p and q be two prepositions given by p : I play cricket during the holidays, q : I just sleep throughout the day then , the compound statement p ∧q is
A. I just sleep throughout the day if and only if I play cricket during the holidays
B. I play cricket during the holidays or I just sleep throughout the day
C. If I play cricket during the holidays , I just sleep throughout the day
D. I play cricket during the holidays and just sleep throughout the day
Answer : Option D
Explaination / Solution:

∧ is replaced by and in verbal form

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Q.4
Let p and q be two prepositions given by p : A parallelogram is a rhombus. q : The diagonals are at right angles. The compound proposition “ A parallelogram is a rhombus iff its diagonals are at right angles “ is represented by
A. p→q
B. p↔q
C. p ∨q
D. p ∨q
Answer : Option B
Explaination / Solution:

iff means a double implication statement so symbolic form has ↔

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Q.5
If the inverse of implication p→q is defined as ∼p→∼q , then the inverse of proposition (p∧∼q)→r is
A. None of these
B. ∼r→∼p∨q
C. (∼p∨q)→∼r
D. r→p∧∼q
Answer : Option C
Explaination / Solution:

as given the rule of inverse in question it becomes ∼(p∧∼q)→∼r =(∼p∨q)→∼r

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Q.6
∼(p∧q) is logically equivalent to
A. ∼p→q
B. ∼p∨∼q
C. ∼p↔∼q
D. ∼p→∼q
Answer : Option B
Explaination / Solution:

∼(p∧q)≡∼p∨∼q De Morgan's law

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Q.7
Which of the following is not a proposition ?
A. Mathematics is interesting
B. 3 is a prime
C. √2 is irrational
D. 5 is an even integer
Answer : Option A
Explaination / Solution:

The above statement is a fact and the other options are mathematical statements which are proposition .

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

(p∧r)∧(∼r∧∼q) since ∼(p→q)≡p∧∼q

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Q.9
The contrapositive of p→(∼q→∼r) is
A. (q∧∼r)→p
B. (∼q∧r)→∼p
C. (q∨∼r)∨p
D. (q∧∼r)→∼p
Answer : Option B
Explaination / Solution:

The contrapositive of p→q is ∼q→∼p

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Q.10
The statement p→(q→p) is equivalent to
A. p→(p∧q)
B. p→(p↔q)
C. p→(p→q)
D. p→(p∨q)
Answer : Option D
Explaination / Solution:

 p q q→p p∨q p→(q→p) p→(p∨q) T T T T T T T F T T T T F T F T T T F F T F T T

Hence they are equivalent

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