130 + Interview Questions in MATHEMATICAL LOGIC IN GENERAL AWARENESS Page 3 InterviewSolution

101.	The dual of ‘(p ∧ t) ∨ (c ∧ ~q)’ where t is a tautology and c is a contradiction, is(A) (p ∨ c) ∧ (t ∨ ~q) (B) (~p ∧ c) ∧ (t ∨ q) (C) (~p ∨ c) ∧ (t ∨ q) (D) (~p ∨ t) ∧ (c ∨ ~q)
Answer» Correct option: (A) (p ∨ c) ∧ (t ∨ ~q)

Discussion

102.	Contrapositive of p → q is (A) q → p (B) ~q → p (C) ~q → ~p (D) q → ~p
Answer» Correct option: (C) ~q → ~p

Discussion

103.	The symbolic form of logic for the following circuit is(A) (p ∨ q) ∧ (~p ∧ r ∨ ~q) ∨ ~r (B) (p ∧ q) ∧ (~p ∨ r ∧ ~q) ∨ ~r (C) (p ∧ q) ∨ [~p ∧ (r ∨ ~q)] ∨ ~r (D) (p ∨ q) ∧ [~p ∨ (r ∧ ~q)] ∨ ~rH
Answer» Correct option: (C) (p ∧ q) ∨ [~p ∧ (r ∨ ~q)] ∨ ~r

Discussion

104.	Which of the following is an incorrect statement in logic ? (A) Multiply the numbers 3 and 10. (B) 3 times 10 is equal to 40. (C) What is the product of 3 and 10? (D) 10 times 3 is equal to 30.
Answer» Correct option: (B) 3 times 10 is equal to 40. ‘Incorrect statement’ means a statement in logic with truth value false. Options (A) and (C) are not statements in logic. Option (D) has truth value True. Option (B) is a statement in logic with truth value false.

104.

Which of the following is an incorrect statement in logic ? (A) Multiply the numbers 3 and 10. (B) 3 times 10 is equal to 40. (C) What is the product of 3 and 10? (D) 10 times 3 is equal to 30.

Answer»

Correct option: (B) 3 times 10 is equal to 40.

‘Incorrect statement’ means a statement in logic with truth value false.

Options (A) and (C) are not statements in logic.

Option (D) has truth value True.

Option (B) is a statement in logic with truth value false.

Discussion

105.	Check whether the following switching circuits are logically equivalent – Justify.(A) (i)(ii)(B) (i) (ii)
Answer» (A) Let p : the switch S₁ is closed q : the switch S₂ is closed r : the switch S₃ is closed (A) The symbolic form of the given switching circuits are : p ∧ (q ∨ r) and (p ∧ q) ∨ (p ∧ r) respectively. By Distributive Law, p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) Hence, The given switching circuits are logically equivalent. (B) The symbolic form of the given switching circuits are : (p ∨ q) ∧ (p ∨ r) and p ∨ (q ∧ r) By Distributive Law, p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) Hence, The given switching circuits are logically equivalent.

105.

Check whether the following switching circuits are logically equivalent – Justify.(A) (i)(ii)(B) (i) (ii)

Answer»

(A) Let p : the switch S₁ is closed

q : the switch S₂ is closed

r : the switch S₃ is closed

(A) The symbolic form of the given switching circuits are :

p ∧ (q ∨ r) and (p ∧ q) ∨ (p ∧ r) respectively.

By Distributive Law,

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Hence,

The given switching circuits are logically equivalent.

(B) The symbolic form of the given switching circuits are :

(p ∨ q) ∧ (p ∨ r) and p ∨ (q ∧ r)

By Distributive Law,

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Hence,

The given switching circuits are logically equivalent.

Discussion

106.	The statement p → (~q) is equivalent to(A) q → p (B) ~q → ~p (C) p → ~q (D) ~q → p
Answer» Correct option: (B) ~q → ~p p → (~q) ≡ ~p ∨ ~q ≡ ~q ∨ ~p

106.

The statement p → (~q) is equivalent to(A) q → p (B) ~q → ~p (C) p → ~q (D) ~q → p

Answer»

Correct option: (B) ~q → ~p

p → (~q) ≡ ~p ∨ ~q

≡ ~q ∨ ~p

Discussion

107.	The negation of p ∨ ( ~ q ∧ ~ p) is (A) ~ p ∧ q (B) p ∧ ~ q (C) ~ p ∨ ~ q (D) ~ p ∨ ~ q
Answer» Correct option: (A) ~ p ∧ q ~[ p ∨ ( ~ q ∧ ~ p)] ≡ ~ p ∧ ~ (~ q ∧ ~ p) ….[By De Morgan’s law] ≡ ~ p ∧ [ ~ ( ~ q ) ∨ ~ (~ p) ] ≡ ~ p ∧ (q ∨ p) ≡ ( ~ p ∧ q ) ∨ ( ~ p ∧ p) ….[Distributive property] ≡ ( ~ p ∧ q ) ∨ F ….[Complement law] ≡ ~ p ∧ q ….[Identity law]

107.

The negation of p ∨ ( ~ q ∧ ~ p) is (A) ~ p ∧ q (B) p ∧ ~ q (C) ~ p ∨ ~ q (D) ~ p ∨ ~ q

Answer»

Correct option: (A) ~ p ∧ q

~[ p ∨ ( ~ q ∧ ~ p)]

≡ ~ p ∧ ~ (~ q ∧ ~ p) ….[By De Morgan’s law]

≡ ~ p ∧ [ ~ ( ~ q ) ∨ ~ (~ p) ]

≡ ~ p ∧ (q ∨ p)

≡ ( ~ p ∧ q ) ∨ ( ~ p ∧ p) ….[Distributive property]

≡ ( ~ p ∧ q ) ∨ F ….[Complement law]

≡ ~ p ∧ q ….[Identity law]

Discussion

108.	The negation of q ∨ ~(p ∧ r) is(A) ~q ∧ ~(p ∨ r) (B) ~q ∧ (p ∧ r) (C) ~q ∧ (p ∧ r) (D) ~q ∧ (p ∧ r)
Answer» Correct option: (B) ~q ∧ (p ∧ r) Negation of q ∨ ~(p ∧ r) is ~[q ∨ ~(p ∧ r)] ≡ ~q ∧ ~(~(p ∧ r)) ≡ ~q ∧ (p ∧ r)

108.

The negation of q ∨ ~(p ∧ r) is(A) ~q ∧ ~(p ∨ r) (B) ~q ∧ (p ∧ r) (C) ~q ∧ (p ∧ r) (D) ~q ∧ (p ∧ r)

Answer»

Correct option: (B) ~q ∧ (p ∧ r)

Negation of q ∨ ~(p ∧ r) is

~[q ∨ ~(p ∧ r)] ≡ ~q ∧ ~(~(p ∧ r))

≡ ~q ∧ (p ∧ r)

Discussion

109.	(p ∧ q) ∨ (~q ∧ p) ≡(A) ∨ p (B) p (C) ~q (D) p ∧ q
Answer» Correct option: (B) p (p ∧ q) ∨ (~q ∧ p) ≡ (p ∧ q) ∨ (p ∧ ~ q) ≡ p ∧ (q ∨ ~q) ≡ p ∧ T ≡ p

109.

(p ∧ q) ∨ (~q ∧ p) ≡(A) ∨ p (B) p (C) ~q (D) p ∧ q

Answer»

Correct option: (B) p

(p ∧ q) ∨ (~q ∧ p) ≡ (p ∧ q) ∨ (p ∧ ~ q)

≡ p ∧ (q ∨ ~q)

≡ p ∧ T ≡ p

Discussion

110.	Which of the following is logically equivalent to ~[p → (p ∨ ~q)]? (A) p ∨ (~p ∧ q ) (B) p ∧ (~p ∧ q) (C) p ∧ (p ∨ ~q) (D) p ∨ (p ∧ ~q)
Answer» Correct option: (B) p ∧ (~p ∧ q) ~[p → (p ∨ ~q)] ≡ p ∧ ~[p ∨ (~q)] ≡ p ∧ (~p ∧ q)

110.

Which of the following is logically equivalent to ~[p → (p ∨ ~q)]? (A) p ∨ (~p ∧ q ) (B) p ∧ (~p ∧ q) (C) p ∧ (p ∨ ~q) (D) p ∨ (p ∧ ~q)

Answer»

Correct option: (B) p ∧ (~p ∧ q)

~[p → (p ∨ ~q)] ≡ p ∧ ~[p ∨ (~q)]

≡ p ∧ (~p ∧ q)

Discussion

111.	Dual of the statement (p ∧ q) ∨ ~q ≡ p ∨ ~q is (A) (p ∨ q) ∨ ~q ∨ p ∨ ~q (B) (p ∧ q) ∧ ~q ∧ p ≡ ~q (C) (p ∧ q) ∧ ~q ∧ p ≡ ~q (D) (~p ∧ ~q) ∧ q ∧ ~p ≡ q
Answer» Correct option: (C) (p ∧ q) ∧ ~q ∧ p ≡ ~q

Discussion

112.	If p and q are true statements in logic, which of the following statement pattern is true?(A) (p ∨ q) → ~ q (B) (p ∨ q) ∧ ~ q (C) (p ∧ ~q) ∧ q (D) (~p ∧ q) ∧ q
Answer» Correct option: (C) (p ∧ ~q) ∧ q (p ∧ ~q) → q ≡ (T ∧ ~T) → T ≡ (T ∧ F) → T ≡ F → T ≡ T

112.

If p and q are true statements in logic, which of the following statement pattern is true?(A) (p ∨ q) → ~ q (B) (p ∨ q) ∧ ~ q (C) (p ∧ ~q) ∧ q (D) (~p ∧ q) ∧ q

Answer»

Correct option: (C) (p ∧ ~q) ∧ q

(p ∧ ~q) → q ≡ (T ∧ ~T) → T

≡ (T ∧ F) → T

≡ F → T

≡ T

Discussion

113.	Which of the following is not true for any two statements p and q?(A) ~[p ∨ (~q)] ≡ ~p ∧ q (B) (p ∨ q) ∨ (~q) is a tautology (C) ~(p ∧ ~p) is a tautology (D) ~(p ∨ q) ≡ ~p ∨ ~q
Answer» Correct option: (D) ~(p ∨ q) ≡ ~p ∨ ~q ~(p ∨ q) ≡ ~p ∨ ~q is not true as it contradicts De Morgan’s law. ∴ option (D) is not true.

113.

Which of the following is not true for any two statements p and q?(A) ~[p ∨ (~q)] ≡ ~p ∧ q (B) (p ∨ q) ∨ (~q) is a tautology (C) ~(p ∧ ~p) is a tautology (D) ~(p ∨ q) ≡ ~p ∨ ~q

Answer»

Correct option: (D) ~(p ∨ q) ≡ ~p ∨ ~q

~(p ∨ q) ≡ ~p ∨ ~q is not true as it contradicts De Morgan’s law.

∴ option (D) is not true.

Discussion

114.	The Boolean Expression (p ∧ ~q) ∨ q ∨ (~p ∧ q) is equivalent to:(A) p ∧ q (B) p ∨ q (C) p ∨ ~q (D) ~p ∧ q
Answer» Correct option: (B) p ∨ q (p ∧ ~q) ∨ q ∨ (~p ∧ q) ≡ [(p ∨ q) ∧ (~q ∨ q)] ∨ (~p ∧ q) ≡ [(p ∨ q) ∨ T] ∨ (~p ∧ q)] ≡ (p ∨ q) ∨ (~p ∨ q) ≡ (T ∨ q) ∧ (p ∨ q) ≡ T ∧ (p ∨ q) p ∨ q

114.

The Boolean Expression (p ∧ ~q) ∨ q ∨ (~p ∧ q) is equivalent to:(A) p ∧ q (B) p ∨ q (C) p ∨ ~q (D) ~p ∧ q

Answer»

Correct option: (B) p ∨ q

(p ∧ ~q) ∨ q ∨ (~p ∧ q)

≡ [(p ∨ q) ∧ (~q ∨ q)] ∨ (~p ∧ q)

≡ [(p ∨ q) ∨ T] ∨ (~p ∧ q)]

≡ (p ∨ q) ∨ (~p ∨ q)

≡ (T ∨ q) ∧ (p ∨ q) ≡ T ∧ (p ∨ q)

p ∨ q

Discussion

115.	Negation of (p ∧ q) → (~ p ∨ r) is(A) (p ∨ q) ∧ (p ∧ ~ r) (B) (p ∧ q) ∨ (p ∧ ~ r)(C) (p ∧ q) ∧ (p ∧ ~ r) (D) (p ∨ q) ∨ (p ∧ ~ r)
Answer» Correct option: (C) (p ∧ q) ∧ (p ∧ ~ r) Since, p → q ≡ ~p ∨ q ∴ ~[(p ∧ q) → (~p ∨ r)] ≡ ~[~(p ∧ q) ∨ (~p ∨ r)] ≡ ~[(~p ∨ ~q) ∨ (~p ∨ r)] ≡ ~(~p ∨ ~q) ∧ ~(~p ∨ r) ≡ (p ∧ q) ∧ (p ∧ ~r)

115.

Negation of (p ∧ q) → (~ p ∨ r) is(A) (p ∨ q) ∧ (p ∧ ~ r) (B) (p ∧ q) ∨ (p ∧ ~ r)(C) (p ∧ q) ∧ (p ∧ ~ r) (D) (p ∨ q) ∨ (p ∧ ~ r)

Answer»

Correct option: (C) (p ∧ q) ∧ (p ∧ ~ r)

Since, p → q ≡ ~p ∨ q

∴ ~[(p ∧ q) → (~p ∨ r)]

≡ ~[~(p ∧ q) ∨ (~p ∨ r)]

≡ ~[(~p ∨ ~q) ∨ (~p ∨ r)]

≡ ~(~p ∨ ~q) ∧ ~(~p ∨ r)

≡ (p ∧ q) ∧ (p ∧ ~r)

Discussion

116.	The logically equivalent statement of p ↔ q is(A) (p ∧ q) ∨ (q → p) (B) (p ∧ q) → ( p ∨ q) (C) (p → q) ∧ (q → p) (D) (p ∧ q) ∨ (p ∧ q)
Answer» Correct option: (C) (p → q) ∧ (q → p)

Discussion

117.

The proposition (p ∧ q) ∧ (p → ~q) is (A) Contradiction (B) Tautology (C) Contingency (D) Tautology and Contradiction

Answer»

Correct option: (A) Contradiction

p	q	~q	p ∧ q	p→ ~q	(p ∧ q) ∧ (p → ~q)
T	T	F	T	F	F
T	F	T	F	T	F
F	T	F	F	T	F
F	F	T	F	T	F

Discussion

118.

~(~p) ↔ p is (A) a tautology (B) a contradiction (C) neither a contradiction nor a tautology (D) none of these

Answer»

Correct option: (A) a tautology

p	~p	~(~p)	~(~p) ↔ p
T	F	T	T
F	T	F	T

All the entries in the last column of the above truth table is T.

∴ ~(~p) ↔ p is a tautology.

Discussion

119.

The proposition (p → q) ↔ ( ~p → ~q) is a (A) tautology (B) contradiction (C) contingency (D) none of these

Answer»

Correct option: (C) contingency

p	q	p → q	~p	~q	~p → ~q	(p → q) ↔ (~p → ~q)
T	T	T	F	F	T	T
T	F	F	F	T	T	F
F	T	T	T	F	F	F
F	F	T	T	T	T	T

Discussion

120.	If p is true and q is false then the truth values of (p → q) ↔ (~q → ~p) and (~p ∨ q) ∧ (~q ∨ p) are respectively (A) F, F (B) F, T (C) T, F (D) T, T
Answer» Correct option: (C) T, F (p → q) ↔ (~q → ~p) and (~p ∨ q) ∧ (~q ∨ p) ∴ (T → F) ↔ (~F → ~T) and (~T ∨ F) ∧ (~F ∨ T) ⇒ F ↔ (T → F) and (F ∨ F) ∧ (T ∨ T) ⇒ F ↔ F and F ∧ T ⇒ T and F

120.

If p is true and q is false then the truth values of (p → q) ↔ (~q → ~p) and (~p ∨ q) ∧ (~q ∨ p) are respectively (A) F, F (B) F, T (C) T, F (D) T, T

Answer»

Correct option: (C) T, F

(p → q) ↔ (~q → ~p) and (~p ∨ q) ∧ (~q ∨ p)

∴ (T → F) ↔ (~F → ~T) and (~T ∨ F) ∧ (~F ∨ T)

⇒ F ↔ (T → F) and (F ∨ F) ∧ (T ∨ T)

⇒ F ↔ F and F ∧ T ⇒ T and F

Discussion

121.	If p and q have truth value ‘F’, then the truth values of (~p ∨ q) ↔ ~(p ∧ q) and ~p ↔ (p → ~q) are respectively(A) T, T (B) F, F (C) T, F (D) F, T
Answer» Correct option: (A) T, T (~p ∨ q) ↔ ~(p ∧ q) and ~p ↔ (p → ~q) ∴ (~F ∨ F) ↔ ~(F ∧ F) and ~F ↔ (F → ~F) ⇒ (T ∨ F) ↔ ~F and T ↔ (F → T) ⇒ T ↔ T and T ↔ T ⇒ T and T

121.

If p and q have truth value ‘F’, then the truth values of (~p ∨ q) ↔ ~(p ∧ q) and ~p ↔ (p → ~q) are respectively(A) T, T (B) F, F (C) T, F (D) F, T

Answer»

Correct option: (A) T, T

(~p ∨ q) ↔ ~(p ∧ q) and ~p ↔ (p → ~q)

∴ (~F ∨ F) ↔ ~(F ∧ F) and ~F ↔ (F → ~F)

⇒ (T ∨ F) ↔ ~F and T ↔ (F → T)

⇒ T ↔ T and T ↔ T

⇒ T and T

Discussion

122.

The statement (p ∧ q) → p is (A) a contradiction (B) a tautology (C) either (A) or (B) (D) a contingency

Answer»

Correct option: (B) a tautology

p	q	p ∧ q	(p ∧ q) → p
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	T

Discussion

123.	If a: Vijay becomes a doctor, b: Ajay is an engineer. Then the statement ‘Vijay becomes a doctor if and only if Ajay is an engineer’ can be written in symbolic form as(A) b ↔ ~a (B) a ↔ b (C) a → b (D) b → a
Answer» Correct option: (B) a ↔ b “if and only if” is expressed as ‘↔’ ∴ symbolic form is a ↔ b.

123.

If a: Vijay becomes a doctor, b: Ajay is an engineer. Then the statement ‘Vijay becomes a doctor if and only if Ajay is an engineer’ can be written in symbolic form as(A) b ↔ ~a (B) a ↔ b (C) a → b (D) b → a

Answer»

Correct option: (B) a ↔ b

“if and only if” is expressed as ‘↔’

∴ symbolic form is a ↔ b.

Discussion

124.	Write the duals of (p ∨ q) ∧ (r ∨ s).
Answer» The duals of the given statement patterns are : (p ∧ q) ∨ (r ∧ s)

124.

Write the duals of (p ∨ q) ∧ (r ∨ s).

Answer»

The duals of the given statement patterns are :

(p ∧ q) ∨ (r ∧ s)

Discussion

125.	If A = {1, 2, 3, 4, 5} then which of the following is not true?(a) Ǝ x ∈ A such that x + 3 = 8(b) Ǝ x ∈ A such that x + 2 < 9(c) Ǝ x ∈ A, x + 6 ≥ 9(d) Ǝ x ∈ A such that x + 6 < 10
Answer» Option : (c) Ǝ x ∈ A, x + 6 ≥ 9

Discussion

126.	Write the truth values of the following statements :√5 is an irrational but 3√5 is a complex number.
Answer» Let p : √5 is an irrational. q : 3√5 is a complex number. Then, The symbolic form of the given statement is p ∧ q. The truth values of p and q are T and F respectively. ∴ The truth value of p ∧ q is F. … [T ∧ F ≡ F]

126.

Write the truth values of the following statements :√5 is an irrational but 3√5 is a complex number.

Answer»

Let p : √5 is an irrational.

q : 3√5 is a complex number.

Then,

The symbolic form of the given statement is p ∧ q.

The truth values of p and q are T and F respectively.

∴ The truth value of p ∧ q is F. … [T ∧ F ≡ F]

Discussion

127.	Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :(i) 4! = 24.(ii) π is an irrational number.(iii) India is a country and Himalayas is a river.(iv) Please get me a glass of water.(v) cos2θ – sin2θ = cos2θ for all θ ∈ R.(vi) If x is a whole number the x + 6 = 0.
Answer» (i) It is a statement which is true, hence its truth value is ‘T’. (ii) It is a statement which is true, hence its truth value is ‘T’. (iii) It is a statement which is false, hence its truth value is ‘F’. ….[T ∧ F ≡ F] (iv) It is an imperative sentence, hence it is not a statement. (v) It is a statement which is true, hence its truth value is ‘T’. (vi) It is a statement which is false, hence its truth value is ‘F’.

127.

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :(i) 4! = 24.(ii) π is an irrational number.(iii) India is a country and Himalayas is a river.(iv) Please get me a glass of water.(v) cos2θ – sin2θ = cos2θ for all θ ∈ R.(vi) If x is a whole number the x + 6 = 0.

Answer»

(i) It is a statement which is true, hence its truth value is ‘T’.

(ii) It is a statement which is true, hence its truth value is ‘T’.

(iii) It is a statement which is false, hence its truth value is ‘F’. ….[T ∧ F ≡ F]

(iv) It is an imperative sentence, hence it is not a statement.

(v) It is a statement which is true, hence its truth value is ‘T’.

(vi) It is a statement which is false, hence its truth value is ‘F’.

Discussion

128.	The negation of ‘If it is Sunday then it is a holiday’ is (A) It is a holiday but not a Sunday. (B) No Sunday then no holiday. (C) It is Sunday, but it is not a holiday, (D) No holiday therefore no Sunday.
Answer» Correct option: (C) It is Sunday, but it is not a holiday, p : It is Sunday q : It is a holiday ∴ Symbolic form p → q ~ (p → q) ≡ p ∧ ~ q i.e. It is Sunday, but it is not a holiday

128.

The negation of ‘If it is Sunday then it is a holiday’ is (A) It is a holiday but not a Sunday. (B) No Sunday then no holiday. (C) It is Sunday, but it is not a holiday, (D) No holiday therefore no Sunday.

Answer»

Correct option: (C) It is Sunday, but it is not a holiday,

p : It is Sunday

q : It is a holiday

∴ Symbolic form p → q

~ (p → q) ≡ p ∧ ~ q

i.e. It is Sunday, but it is not a holiday

Discussion

129.	The negation of ‘For every natural number x, x + 5 > 4’ is (A) ∀ x ∈ N, x + 5 < 4 (B) ∀ x ∈ N, x - 5 < 4 (C) For every integer x, x + 5 < 4 (D) There exists a natural number x, for which x + 5 ≤ 4
Answer» Correct option: (D) There exists a natural number x, for which x + 5 ≤ 4 Given statement is ‘∀ x ∈ N, x + 5 > 4’ ∴ ~ [ ∀ x ∈ N, , x + 5 > 4] ≡ ∃ x ∈ N, such that x + 5 ≤ 4 i.e., there exists a natural number x, for which x + 5 ≤ 4

129.

The negation of ‘For every natural number x, x + 5 > 4’ is (A) ∀ x ∈ N, x + 5 < 4 (B) ∀ x ∈ N, x - 5 < 4 (C) For every integer x, x + 5 < 4 (D) There exists a natural number x, for which x + 5 ≤ 4

Answer»

Correct option: (D) There exists a natural number x, for which x + 5 ≤ 4

Given statement is ‘∀ x ∈ N, x + 5 > 4’

∴ ~ [ ∀ x ∈ N, , x + 5 > 4]

≡ ∃ x ∈ N, such that x + 5 ≤ 4

i.e., there exists a natural number x, for which x + 5 ≤ 4

Discussion

130.	If p : The examinations are approaching,q : Students study hard, give a verbal statement for each of the following:i. p \(\land\) \(\sim\)qii. p \(\leftrightarrow\) qiii. \(\sim\)p \(\rightarrow\) qiv. p \(\lor\) qv. \(\sim\)q \(\rightarrow\) \(\sim\)p
Answer» i. The examinations are approaching but the students do not study hard. ii. The examinations are approaching if and only if the students study hard. iii. If the examinations are not approaching, then the students study hard. iv. The examinations are approaching or the students study hard. v. If the students do not study hard then the examinations are not approaching.

130.

If p : The examinations are approaching,q : Students study hard, give a verbal statement for each of the following:i. p \(\land\) \(\sim\)qii. p \(\leftrightarrow\) qiii. \(\sim\)p \(\rightarrow\) qiv. p \(\lor\) qv. \(\sim\)q \(\rightarrow\) \(\sim\)p

Answer»

i. The examinations are approaching but the students do not study hard.

ii. The examinations are approaching if and only if the students study hard.

iii. If the examinations are not approaching, then the students study hard.

iv. The examinations are approaching or the students study hard.

v. If the students do not study hard then the examinations are not approaching.

Discussion

Explore topic-wise InterviewSolutions in .

The dual of ‘(p ∧ t) ∨ (c ∧ ~q)’ where t is a tautology and c is a contradiction, is(A) (p ∨ c) ∧ (t ∨ ~q) (B) (~p ∧ c) ∧ (t ∨ q) (C) (~p ∨ c) ∧ (t ∨ q) (D) (~p ∨ t) ∧ (c ∨ ~q)

Contrapositive of p → q is (A) q → p (B) ~q → p (C) ~q → ~p (D) q → ~p

The symbolic form of logic for the following circuit is(A) (p ∨ q) ∧ (~p ∧ r ∨ ~q) ∨ ~r (B) (p ∧ q) ∧ (~p ∨ r ∧ ~q) ∨ ~r (C) (p ∧ q) ∨ [~p ∧ (r ∨ ~q)] ∨ ~r (D) (p ∨ q) ∧ [~p ∨ (r ∧ ~q)] ∨ ~rH

Which of the following is an incorrect statement in logic ? (A) Multiply the numbers 3 and 10. (B) 3 times 10 is equal to 40. (C) What is the product of 3 and 10? (D) 10 times 3 is equal to 30.

Check whether the following switching circuits are logically equivalent – Justify.(A) (i)(ii)(B) (i) (ii)

The statement p → (~q) is equivalent to(A) q → p (B) ~q → ~p (C) p → ~q (D) ~q → p

The negation of p ∨ ( ~ q ∧ ~ p) is (A) ~ p ∧ q (B) p ∧ ~ q (C) ~ p ∨ ~ q (D) ~ p ∨ ~ q

The negation of q ∨ ~(p ∧ r) is(A) ~q ∧ ~(p ∨ r) (B) ~q ∧ (p ∧ r) (C) ~q ∧ (p ∧ r) (D) ~q ∧ (p ∧ r)

(p ∧ q) ∨ (~q ∧ p) ≡(A) ∨ p (B) p (C) ~q (D) p ∧ q

Which of the following is logically equivalent to ~[p → (p ∨ ~q)]? (A) p ∨ (~p ∧ q ) (B) p ∧ (~p ∧ q) (C) p ∧ (p ∨ ~q) (D) p ∨ (p ∧ ~q)

Dual of the statement (p ∧ q) ∨ ~q ≡ p ∨ ~q is (A) (p ∨ q) ∨ ~q ∨ p ∨ ~q (B) (p ∧ q) ∧ ~q ∧ p ≡ ~q (C) (p ∧ q) ∧ ~q ∧ p ≡ ~q (D) (~p ∧ ~q) ∧ q ∧ ~p ≡ q

If p and q are true statements in logic, which of the following statement pattern is true?(A) (p ∨ q) → ~ q (B) (p ∨ q) ∧ ~ q (C) (p ∧ ~q) ∧ q (D) (~p ∧ q) ∧ q

Which of the following is not true for any two statements p and q?(A) ~[p ∨ (~q)] ≡ ~p ∧ q (B) (p ∨ q) ∨ (~q) is a tautology (C) ~(p ∧ ~p) is a tautology (D) ~(p ∨ q) ≡ ~p ∨ ~q

The Boolean Expression (p ∧ ~q) ∨ q ∨ (~p ∧ q) is equivalent to:(A) p ∧ q (B) p ∨ q (C) p ∨ ~q (D) ~p ∧ q

Negation of (p ∧ q) → (~ p ∨ r) is(A) (p ∨ q) ∧ (p ∧ ~ r) (B) (p ∧ q) ∨ (p ∧ ~ r)(C) (p ∧ q) ∧ (p ∧ ~ r) (D) (p ∨ q) ∨ (p ∧ ~ r)

The logically equivalent statement of p ↔ q is(A) (p ∧ q) ∨ (q → p) (B) (p ∧ q) → ( p ∨ q) (C) (p → q) ∧ (q → p) (D) (p ∧ q) ∨ (p ∧ q)

The proposition (p ∧ q) ∧ (p → ~q) is (A) Contradiction (B) Tautology (C) Contingency (D) Tautology and Contradiction

~(~p) ↔ p is (A) a tautology (B) a contradiction (C) neither a contradiction nor a tautology (D) none of these

The proposition (p → q) ↔ ( ~p → ~q) is a (A) tautology (B) contradiction (C) contingency (D) none of these

If p is true and q is false then the truth values of (p → q) ↔ (~q → ~p) and (~p ∨ q) ∧ (~q ∨ p) are respectively (A) F, F (B) F, T (C) T, F (D) T, T

If p and q have truth value ‘F’, then the truth values of (~p ∨ q) ↔ ~(p ∧ q) and ~p ↔ (p → ~q) are respectively(A) T, T (B) F, F (C) T, F (D) F, T

The statement (p ∧ q) → p is (A) a contradiction (B) a tautology (C) either (A) or (B) (D) a contingency

If a: Vijay becomes a doctor, b: Ajay is an engineer. Then the statement ‘Vijay becomes a doctor if and only if Ajay is an engineer’ can be written in symbolic form as(A) b ↔ ~a (B) a ↔ b (C) a → b (D) b → a

Write the duals of (p ∨ q) ∧ (r ∨ s).

If A = {1, 2, 3, 4, 5} then which of the following is not true?(a) Ǝ x ∈ A such that x + 3 = 8(b) Ǝ x ∈ A such that x + 2 &lt; 9(c) Ǝ x ∈ A, x + 6 ≥ 9(d) Ǝ x ∈ A such that x + 6 &lt; 10

Write the truth values of the following statements :√5 is an irrational but 3√5 is a complex number.

The negation of ‘If it is Sunday then it is a holiday’ is (A) It is a holiday but not a Sunday. (B) No Sunday then no holiday. (C) It is Sunday, but it is not a holiday, (D) No holiday therefore no Sunday.

The negation of ‘For every natural number x, x + 5 &gt; 4’ is (A) ∀ x ∈ N, x + 5 &lt; 4 (B) ∀ x ∈ N, x - 5 &lt; 4 (C) For every integer x, x + 5 &lt; 4 (D) There exists a natural number x, for which x + 5 ≤ 4

If p : The examinations are approaching,q : Students study hard, give a verbal statement for each of the following:i. p \(\land\) \(\sim\)qii. p \(\leftrightarrow\) qiii. \(\sim\)p \(\rightarrow\) qiv. p \(\lor\) qv. \(\sim\)q \(\rightarrow\) \(\sim\)p

If A = {1, 2, 3, 4, 5} then which of the following is not true?(a) Ǝ x ∈ A such that x + 3 = 8(b) Ǝ x ∈ A such that x + 2 < 9(c) Ǝ x ∈ A, x + 6 ≥ 9(d) Ǝ x ∈ A such that x + 6 < 10

The negation of ‘For every natural number x, x + 5 > 4’ is (A) ∀ x ∈ N, x + 5 < 4 (B) ∀ x ∈ N, x - 5 < 4 (C) For every integer x, x + 5 < 4 (D) There exists a natural number x, for which x + 5 ≤ 4