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

**Answer : ****Option C**

**Explaination / Solution: **

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

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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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

**Answer : ****Option D**

**Explaination / Solution: **

using conjunction to combine statement

using conjunction to combine statement

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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

**Answer : ****Option A**

**Explaination / Solution: **

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

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

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∼(p∨q)∨(∼p∧q) is logically equivalent to

**A. ** ∼p

**B. ** p

**C. ** q

**D. ** ∼q

**Answer : ****Option A**

**Explaination / Solution: **

distributive law

since

since

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∼p∨∼q is logically equivalent to

**A. ** ∼p→∼q

**B. ** p→∼q

**C. ** p→∼q

**D. ** p↔q

**Answer : ****Option B**

**Explaination / Solution: **

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

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

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The false statement in the following is

**A. ** p∧∼p is a contradiction

**B. ** ∼(∼p)↔p is a tautology

**C. ** p∨∼p is a tautology

**D. ** (p→q)↔(∼q→∼p) is a contradiction

**Answer : ****Option D**

**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.

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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Negation of the statement p→(q∧r) is

**A. ** ∼p→(∼q∨r)

**B. ** p→(∼q∨r)

**C. ** ∼p∧(∼q∨∼r)

**D. ** p∧(∼q∨∼r)

**Answer : ****Option D**

**Explaination / Solution: **

= since

=

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The contrapositive of (p∨q)→r is

**A. ** p→(p∧q)

**B. ** ∼r→(∼p∧∼q)

**C. ** ∼r→(p∧q)

**D. ** ∼r→∼(p∨q)

**Answer : ****Option D**

**Explaination / Solution: **

the contrapositive of p→q is∼q→∼p

the contrapositive of p→q is∼q→∼p

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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)

**Answer : ****Option D**

**Explaination / Solution: **

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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

**Answer : ****Option A**

**Explaination / Solution: **

The contrapositive of p→q≡∼q→∼p

The contrapositive of p→q≡∼q→∼p

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