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1.

Negation of the statement p → (q ∧ r) is1. ~ p → ~ (q ∧r)2. ~ p ∨(q ∧r)3. (q ∧r) → p4. p ∧(~ q ∨ ~ r)

Answer» Correct Answer - Option 4 : p ∧(~ q ∨ ~ r)

p ∧(~ (q ∧r))

= p ∧(~ q ∨~ r)
Discussion
2.

Which one of the following statements is not a tautology?1. (p ∨ q) → (p ∨ (~q))2. (p ∧ q) → (~p) ∨ q3. p → (p ∨ q)4. (p ∧ q) → p

Answer» Correct Answer - Option 2 : (p ∧ q) → (~p) ∨ q

p

q

~q

p ∨ ~q

~p

p ∧ ~q

T

T

F

T

F

F

T

F

T

T

F

T

F

T

F

F

T

F

F

F

T

T

T

F

p

q

p ∨ q

p → p ∨ q

p ∧ q

p ∧ q → p

T

T

T

T

T

T

T

F

T

T

F

T

F

T

T

T

F

T

F

F

F

T

F

T

p

q

~p ∨ q

(p ∧ q) → (~p) ∨ q

(p ∨ q) → (p ∨ (~q))

T

T

T

T

T

T

F

F

T

T

F

T

T

T

F

F

F

T

T

T

A tautology is a formula or assertion that is true in every possible interpretation. So, by the truth table (p ∨ q) → (p ∨ (~q)) statement is not a tautology.

Atautologyis a statement that is alwaystrue, no matter what. If you construct atruth tablefor a statement and all of the column values for the statement aretrue(T), then the statement is a tautologybecause it's alwaystrue!
Discussion
3.

The contrapositive of the statement "If you are born in India, then you are a citizen of India" is1. If you are not a citizen of India, then you are not born in India.2. If you are born in India, then you are not a citizen of India.3. If you are not born in India, then you are a citizen of India.4. If you are a citizen of India, then you are born in India.

Answer» Correct Answer - Option 1 : If you are not a citizen of India, then you are not born in India.

Now, “If you are born in India” is p

“you are a citizen of India" is q

It is known that contrapositive ofp ⇒ qis~q ⇒ ~p

“if you are not a citizen of India” is ~q

“then you are not born in India” is ~p

Thus, ~q ⇒ ~pis “If you are not a citizen of India, then you are not born in India”

Thus, option (a) is the correct answer.
Discussion
4.

Which of the following symbols should be placed in the blank spaces respectively (in the same order from left to right) in order to complete the expression in such a manner that Q> Pholds definitely true?S _ P _U _ T = Q _ R1. >,≤,≤,≥2. >,≤, >,≥3. >,≤, =,≥4. >,≤,<,≥

Answer» Correct Answer - Option 4 : >,≤,<,≥

Let us check each option:-

1) S > P≤ U≤ T = Q≥ R→False (as P≤ U≤ T = Q → Q≥ P)

2) S > P≤ U> T = Q≥ R→ False(asP≤ U> T = Q→Thus clear relation between Qand P cannot be determined)

3) S > P≤ U= T = Q≥ R→ False(asP≤ U= T = Q→Q≥P)

4) S > P≤ U< T= Q ≥ R→ True(asP≤ U< T= Q→Q> P)

Hence,>,≤,<,≥ is the correct answer.
Discussion