130 + Interview Questions in MATHEMATICAL LOGIC IN GENERAL AWARENESS Page 2 InterviewSolution

51.	A compound statement p or q is false only when (A) p is false. (B) q is false. (C) both p and q are false. (D) depends on p and q.
Answer» Correct option: (C) both p and q are false.

Discussion

52.	Assuming p: She is beautiful,q: She is clever, the verbal form of p\(\land\) (~q) is(A) She is beautiful but not clever.(B) She is beautiful and clever.(C) She is not beautiful and not clever.(D) She is beautiful or not clever.
Answer» Correct answer: (A) She is beautiful but not clever.

Discussion

53.	Using the statementsp: Kiran passed the examination,s : Kiran is sad.the statement ‘It is not true that Kiran passestherefore he is sad’ in symbolic form is(A) ~p \(\rightarrow\) s (B) ~ (p \(\rightarrow\) ~ s) (C) ~p \(\rightarrow\) ~ s (D) ~ (p \(\rightarrow\) s)
Answer» Correct answer: (D) ~ (p \(\rightarrow\)s)

Discussion

54.	If p: Ram is lazy, q: Ram fails in the examination, then the verbal form of ~p ∨ ~q is (A) Ram is not lazy and he fails in the examination. (B) Ram is not lazy or he does not fail in the examination. (C) Ram is lazy or he does not fail in the examination. (D) Ram is not lazy and he does not fail in the examination.
Answer» Correct option: (B) Ram is not lazy or he does not fail in the examination. ~p: Ram is not lazy, ~q: Ram does not fail in the examination, ‘∨’ indicates ‘or’.

54.

If p: Ram is lazy, q: Ram fails in the examination, then the verbal form of ~p ∨ ~q is (A) Ram is not lazy and he fails in the examination. (B) Ram is not lazy or he does not fail in the examination. (C) Ram is lazy or he does not fail in the examination. (D) Ram is not lazy and he does not fail in the examination.

Answer»

Correct option: (B) Ram is not lazy or he does not fail in the examination.

~p: Ram is not lazy, ~q: Ram does not fail in the examination, ‘∨’ indicates ‘or’.

Discussion

55.	If p: Sita gets promotion, q: Sita is transferred to Pune. The verbal form of ~p ↔ q is written as (A) Sita gets promotion and Sita gets transferred to Pune. (B) Sita does not get promotion then Sita will be transferred to Pune. (C) Sita gets promotion if Sita is transferred to Pune. (D) Sita does not get promotion if and only if Sita is transferred to Pune.
Answer» Correct option: (D) Sita does not get promotion if and only if Sita is transferred to Pune. ~p: Sita does not get promotion and ‘↔’ symbol indicates ‘if and only if’.

55.

If p: Sita gets promotion, q: Sita is transferred to Pune. The verbal form of ~p ↔ q is written as (A) Sita gets promotion and Sita gets transferred to Pune. (B) Sita does not get promotion then Sita will be transferred to Pune. (C) Sita gets promotion if Sita is transferred to Pune. (D) Sita does not get promotion if and only if Sita is transferred to Pune.

Answer»

Correct option: (D) Sita does not get promotion if and only if Sita is transferred to Pune.

~p: Sita does not get promotion and ‘↔’ symbol indicates ‘if and only if’.

Discussion

56.	Assuming the first part of each statement as p, second as q and the third as r, the statement ‘If A, B, C are three distinct points, then either they are collinear or they form a triangle’ in symbolic form is (A) p ↔ (q ∨ r) (B) (p ∧ q) → r (C) p → (q ∨ r) (D) p → (q ∧ r)
Answer» Correct option: (C) p → (q ∨ r) p: A, B,C, are distinct points q: Points are collinear r: Points form a triangle ∴ p implies (q or r) i.e. p → (q ∨ r)

56.

Assuming the first part of each statement as p, second as q and the third as r, the statement ‘If A, B, C are three distinct points, then either they are collinear or they form a triangle’ in symbolic form is (A) p ↔ (q ∨ r) (B) (p ∧ q) → r (C) p → (q ∨ r) (D) p → (q ∧ r)

Answer»

Correct option: (C) p → (q ∨ r)

p: A, B,C, are distinct points

q: Points are collinear

r: Points form a triangle

∴ p implies (q or r) i.e. p → (q ∨ r)

Discussion

57.	Assuming p: She is beautiful, q: She is clever,the verbal form of ~p \(\land\) (~q) is(A) She is beautiful but not clever.(B) She is beautiful and clever.(C) She is not beautiful and not clever.(D) She is beautiful or not clever.
Answer» Correct answer: (C) She is not beautiful and not clever.

Discussion

58.	Let p: ‘It is hot’ andq: ‘It is raining’.The verbal statement for (p \(\land\) ~q) \(\rightarrow\) p is(A) If it is hot and not raining, then it is hot.(B) If it is hot and raining, then it is hot.(C) If it is hot or raining, then it is not hot.(D) If it is hot and raining, then it is not hot.
Answer» Correct answer: (A) If it is hot and not raining, then it is hot.

Discussion

59.	If A = {2, 3, 4, 5, 6}, then which of the following is not true?(A) \(\exists\) x \(\in\) A such that x + 3 = 8(B) \(\exists\) x \(\in\) A such that x + 2 < 5(C) \(\exists\) x \(\in\) A such that x + 2 < 9(D) \(\forall\) x \(\in\) A such that x + 6 \(\geq\) 9
Answer» Correct answer: (D) \(\forall\) x \(\in\) A such that x + 6 \(\geq\) 9

Discussion

60.	If p and q be two statements then the conjunction of the statements, p \(\land\) q is false when(A) both p and q are true. (B) either p or q are true(C) either p or q or both are false. (D) both p and q are false.
Answer» Correct answer: (C) either p or q or both are false.

Discussion

61.	When two statements are connected by the connective ‘if and only if’ then the compound statement is called(A) conjunction of the statements.(B) disjunction of the statements.(C) biconditional statement.(D) conditional statement.
Answer» Correct answer: (C) biconditional statement.

Discussion

62.	(i) If the compound proposition “(p → q) ∧ (p ∧ r)” is false, then find the truth values of p, q and r. (ii) If the compound proposition p→ (q ∨ r) is false, then find the truth values of p, q and r. (iii) If the compound proposition p → (~q ∨ r) is false, then find the truth values of p, q and r. (iv) If the truth value of the propositions (p ∧ q) → (r ∨ ~s) is false, then find the truth values of p, q, rand s.
Answer» (i) Given (p → q) ∧ (p ∧ r) is false (a) Case 1: p → q is true & p ∧ r is false p is T q is T & p is T &ris F p = T, q = T, r = F Case 2(a): p = F, q = T p ∧ r= F → p = F, r = F p = F, q = T, r = f (b): (p →q) is F & par is true T → F = F T ∧ T is T P=T, q = F, r=T Case 3: (p → q) is F & (p ∧ r) is false T → F = F F ∧ F = F F ∧ T = F F ∧ F = F . ∴ p = T, q = F, r= F. (ii) Given p → (q ∨ r) is false T → F = F ∴ p = T & q ∨ r is false = F ∨ F= F ∴ p = T, q = F & r= F (iii) Given p → (q ∨ r) is false Then T → F= F ∴ P = T, ~ q ∨ r= F F ∨ F = F ∴ q = T, q = T, r = F. (iv) Given (p ^ q) → (r ∨ ~s) is false We know that T → F = F ∴ p∧q = T and r ∨ ~s = F is false T∧ T = T F ∨ F= F is false ∴ p = T, q = T, r = F, S = T

62.

(i) If the compound proposition “(p → q) ∧ (p ∧ r)” is false, then find the truth values of p, q and r. (ii) If the compound proposition p→ (q ∨ r) is false, then find the truth values of p, q and r. (iii) If the compound proposition p → (~q ∨ r) is false, then find the truth values of p, q and r. (iv) If the truth value of the propositions (p ∧ q) → (r ∨ ~s) is false, then find the truth values of p, q, rand s.

Answer»

(i) Given (p → q) ∧ (p ∧ r) is false

(a) Case 1: p → q is true & p ∧ r is false

p is T q is T & p is T &ris F

p = T, q = T, r = F

Case 2(a): p = F, q = T p ∧ r= F → p = F, r = F

p = F, q = T, r = f

(b): (p →q) is F & par is true

T → F = F T ∧ T is T

P=T, q = F, r=T

Case 3: (p → q) is F & (p ∧ r) is false

T → F = F

F ∧ F = F

F ∧ T = F

F ∧ F = F .

∴ p = T, q = F, r= F.

(ii) Given p → (q ∨ r) is false

T → F = F

∴ p = T & q ∨ r is false

= F ∨ F= F

∴ p = T, q = F & r= F

(iii) Given p → (q ∨ r) is false

Then T → F= F

∴ P = T, ~ q ∨ r= F

F ∨ F = F

∴ q = T, q = T, r = F.

(iv) Given (p ^ q) → (r ∨ ~s) is false

We know that T → F = F

∴ p∧q = T and r ∨ ~s = F is false

T∧ T = T

F ∨ F= F is false

∴ p = T, q = T, r = F, S = T

Discussion

63.	Negation of ~(p \(\lor\) q) is(A) ~p \(\lor\) ~q(B) ~p \(\land\) ~q(C) p \(\land\) ~q(D) p \(\lor\) ~q
Answer» Correct answer: (B) ~p \(\land\) ~q

Discussion

64.	The symbolic form of the following circuit, where p: switch S1 is closed.and q: switch S2 is closed, is-(A) (p\(\lor\)q) \(\land\) [\(\sim\)p \(\lor\)(p\(\land\)~q)](B) (~p\(\land\)q) \(\lor\) [~p\(\lor\) (p\(\land\)~q)](C) (p \(\lor\)q) \(\lor\) [~p \(\land\)(p \(\lor\)~q)](D) (p\(\land\)q) \(\lor\) [~p \(\land\)(p\(\land\)~q)]
Answer» Correct answer: (C) (p \(\lor\)q) \(\lor\) [~p \(\land\)(p \(\lor\)~q)]

Discussion

65.	The dual of ~(p\(\lor\)q) \(\lor\)[p\(\lor\)(q\(\land\)~r)] is, (A) ~(p\(\land\)q) \(\land\) [p \(\lor\) (q\(\land\)~r)](B) (p\(\land\)q) \(\land\) [p \(\land\) (q \(\lor\) ~ r)](C) ~(p\(\land\)q) \(\land\) [p \(\land\) (q\(\land\) r)](D) ~(p\(\land\)q) \(\land\) [p \(\land\) (q \(\lor\) ~ r)]
Answer» Correct answer: (D) ~(p\(\land\)q) \(\land\) [p \(\land\) (q \(\lor\) ~ r)]

Discussion

66.	Using rules in logic, prove the following : (i) p ↔ q ≡ ~ (p ∧ ~q) ∧ ~(q ∧ ~p)(ii) ~p ∧ q ≡ (p ∨ q) ∧ ~p(iii) ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p
Answer» (i) p ↔ q ≡ ~ (p ∧ ~q) ∧ ~(q ∧ ~p) By the rules of negation of biconditional, ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p) ∴ ~ [(p ∧ ~ q) ∨ (q ∧ ~p)] ≡ p ↔ q ∴ ~(p ∧ ~q) ∧ ~(q ∧ ~p) ≡ p ↔ q …(Negation of disjunction) ≡ p ↔ q ≡ ~(p ∧ ~ q) ∧ ~ (q ∧ ~p). (ii) ~p ∧ q ≡ (p ∨ q) ∧ ~p (p ∨ q) ∧ ~ p ≡ (p ∧ ~p) ∨ (q ∧ ~p) … (Distributive Law) ≡ F ∨ (q ∧ ~p) … (Complement Law) ≡ q ∧ ~ p … (Identity Law) ≡ ~p ∧ q …(Commutative Law) ∴ ~p ∧ q ≡ (p ∨ q) ∧ ~p. (iii) ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p ~ (p ∨ q) ∨ (~p ∧ q) ≡ (~p ∧ ~q) ∨ (~p ∧ q) … (Negation of disjunction) ≡ ~p ∧ (~q ∨ q) … (Distributive Law) ≡ ~ p ∧ T … (Complement Law) ≡ ~ p … (Identity Law) ∴ ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p.

66.

Using rules in logic, prove the following : (i) p ↔ q ≡ ~ (p ∧ ~q) ∧ ~(q ∧ ~p)(ii) ~p ∧ q ≡ (p ∨ q) ∧ ~p(iii) ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p

Answer»

(i) p ↔ q ≡ ~ (p ∧ ~q) ∧ ~(q ∧ ~p)

By the rules of negation of biconditional,

~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p)

∴ ~ [(p ∧ ~ q) ∨ (q ∧ ~p)] ≡ p ↔ q

∴ ~(p ∧ ~q) ∧ ~(q ∧ ~p) ≡ p ↔ q …(Negation of disjunction)

≡ p ↔ q ≡ ~(p ∧ ~ q) ∧ ~ (q ∧ ~p).

(ii) ~p ∧ q ≡ (p ∨ q) ∧ ~p

(p ∨ q) ∧ ~ p

≡ (p ∧ ~p) ∨ (q ∧ ~p) … (Distributive Law)

≡ F ∨ (q ∧ ~p) … (Complement Law)

≡ q ∧ ~ p … (Identity Law)

≡ ~p ∧ q …(Commutative Law)

∴ ~p ∧ q ≡ (p ∨ q) ∧ ~p.

(iii) ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p

~ (p ∨ q) ∨ (~p ∧ q)

≡ (~p ∧ ~q) ∨ (~p ∧ q) … (Negation of disjunction)

≡ ~p ∧ (~q ∨ q) … (Distributive Law)

≡ ~ p ∧ T … (Complement Law)

≡ ~ p … (Identity Law)

∴ ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p.

Discussion

67.	Given that p is ‘false’ and q is ‘true’ then the statement which is ‘false’ is(A) ~p → ~q(B) p → (q ∧ p) (C) p → ~q (D) q → ~p
Answer» Correct option: (A) ~p → ~q ~ p → ~ q ≡ ~ F → ~ T ≡ T → F ≡ F p → (q ∧ p) ≡ F → (T ∧ F) ≡ F → F ≡ T p → ~ q ≡ F → ~ T ≡ F → F ≡ T q → ~ p ≡ T → ~ F ≡ T → T ≡ T

67.

Given that p is ‘false’ and q is ‘true’ then the statement which is ‘false’ is(A) ~p → ~q(B) p → (q ∧ p) (C) p → ~q (D) q → ~p

Answer»

Correct option: (A) ~p → ~q

~ p → ~ q ≡ ~ F → ~ T ≡ T → F ≡ F

p → (q ∧ p) ≡ F → (T ∧ F) ≡ F → F ≡ T

p → ~ q ≡ F → ~ T ≡ F → F ≡ T

q → ~ p ≡ T → ~ F ≡ T → T ≡ T

Discussion

68.	If p, q are true and r is false statement then which of the following is true statement? (A) (p ∧ q) ∨ r is F (B) (p ∧ q) → r is T (C) (p ∨ q) ∧ (p ∨ r) is T (D) (p → q) ↔ (p → r) is T
Answer» Correct option: (C) (p ∨ q) ∧ (p ∨ r) is T (p ∨ q) ∧ (p ∨ r) ≡ (T ∨ T) ∧ (T ∨ F) ≡ T ∧ T ≡ T

68.

If p, q are true and r is false statement then which of the following is true statement? (A) (p ∧ q) ∨ r is F (B) (p ∧ q) → r is T (C) (p ∨ q) ∧ (p ∨ r) is T (D) (p → q) ↔ (p → r) is T

Answer»

Correct option: (C) (p ∨ q) ∧ (p ∨ r) is T

(p ∨ q) ∧ (p ∨ r) ≡ (T ∨ T) ∧ (T ∨ F)

≡ T ∧ T

≡ T

Discussion

69.

f ~q ∨ p is F, then which of the following is correct?(A) p ↔ q is T (B) p → q is T (C) q → p is T (D) p → q is F

Answer»

Correct option: (B) p → q is T

p	q	~q	~q ∨ p	p ↔ q	p → q	q → p
T T F F	T F T F	F T F T	T T F T	T F F T	T F T T	T T F T

Alternate Method:

~ q ∨ p: F

∴ ~ q is F, p is F

i.e., q is T, p is F

∴ p → q ≡ F → T ≡ T

Discussion

70.	Find out which of the following statements have the same meaning: i. If Seema solves a problem then she is happy. ii. If Seema does not solve a problem then she is not happy. iii. If Seema is not happy then she hasn’t solved the problem. iv. If Seema is happy then she has solved the problem(A) (i, ii) and (iii, iv) (B) i, ii, iii (C) (i, iii) and (ii, iv) (D) ii, iii, iv
Answer» Correct option: (C) (i, iii) and (ii, iv) p: Seema solves a problem q: She is happy i. p → q ii. ~p → ~q iii. ~q → ~p iv. q → p (i) and (iii) have the same meaning, (ii) and (iv) have the same meaning.

70.

Find out which of the following statements have the same meaning: i. If Seema solves a problem then she is happy. ii. If Seema does not solve a problem then she is not happy. iii. If Seema is not happy then she hasn’t solved the problem. iv. If Seema is happy then she has solved the problem(A) (i, ii) and (iii, iv) (B) i, ii, iii (C) (i, iii) and (ii, iv) (D) ii, iii, iv

Answer»

Correct option: (C) (i, iii) and (ii, iv)

p: Seema solves a problem

q: She is happy

i. p → q

ii. ~p → ~q

iii. ~q → ~p

iv. q → p

(i) and (iii) have the same meaning,

(ii) and (iv) have the same meaning.

Discussion

71.	Find which of the following statements convey the same meanings? i. If it is the bride’s dress then it has to be red. ii. If it is not bride’s dress then it cannot be red. iii. If it is a red dress then it must be the bride’s dress. iv. If it is not a red dress then it can’t be the bride’s dress. (A) (i, iv) and (ii, iii) (B) (i, ii) and (iii, iv) (C) (i), (ii), (iii) (D) (i, iii) and (ii, iv)
Answer» Correct option: (A) (i, iv) and (ii, iii) i. b → r ii. ~b → ~r iii. r → b iv. ~r → ~b (i) and (iv) are the same and (ii) and (iii) are the same.

71.

Find which of the following statements convey the same meanings? i. If it is the bride’s dress then it has to be red. ii. If it is not bride’s dress then it cannot be red. iii. If it is a red dress then it must be the bride’s dress. iv. If it is not a red dress then it can’t be the bride’s dress. (A) (i, iv) and (ii, iii) (B) (i, ii) and (iii, iv) (C) (i), (ii), (iii) (D) (i, iii) and (ii, iv)

Answer»

Correct option: (A) (i, iv) and (ii, iii)

i. b → r

ii. ~b → ~r

iii. r → b

iv. ~r → ~b

(i) and (iv) are the same and (ii) and (iii) are the same.

Discussion

72.	~ (p ∨ q) ∨ (~p ∧ q) is logically equivalent to (A) ~p (B) p (C) q (D) ~q
Answer» Correct option: (A) ~p ~ (p ∨ q) ∨ (~p ∧ q) ≡ (~p ∧ ~q) ∨ (~p ∧ q) ≡ ~p ∧(~q ∨ q) ≡ ~p ∧ T ≡ ~p

72.

~ (p ∨ q) ∨ (~p ∧ q) is logically equivalent to (A) ~p (B) p (C) q (D) ~q

Answer»

Correct option: (A) ~p

~ (p ∨ q) ∨ (~p ∧ q)

≡ (~p ∧ ~q) ∨ (~p ∧ q)

≡ ~p ∧(~q ∨ q)

≡ ~p ∧ T

≡ ~p

Discussion

73.	p ∧ (p → q) is logically equivalent to (A) p ∨ q (B) ~p ∨ q (C) p ∧ q (D) p ∨ ~q
Answer» Correct option: (C) p ∧ q p ∧ (p → q) ≡ p ∧ (~p ∨ q) ….[Conditional law] ≡ (p ∧ ~p) ∨ (p ∧ q) ….[Distributive law] ≡ F ∨ (p ∧ q) ….[Complement law] p ∧ q ….[Identity law]

73.

p ∧ (p → q) is logically equivalent to (A) p ∨ q (B) ~p ∨ q (C) p ∧ q (D) p ∨ ~q

Answer»

Correct option: (C) p ∧ q

p ∧ (p → q)

≡ p ∧ (~p ∨ q) ….[Conditional law]

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

≡ F ∨ (p ∧ q) ….[Complement law]

p ∧ q ….[Identity law]

Discussion

74.	Negation of a statement in logic corresponds to ........... in set theory. (A) empty set (B) null set (C) complement of a set (D) universal set
Answer» Correct option: (C) complement of a set

Discussion

75.	If p and q are two logical statements and A and B are two sets, then p → q corresponds to (A) A ⊆ B (B) A ∩ B (C) A ∪ B (D) A \(\nsubseteq\) B
Answer» Correct option: (A) A ⊆ B

Discussion

76.	The logical statement ‘p ∧ q’ can be related to the set theory’s concept of (A) union of two sets (B) intersection of two set (C) subset of a set (D) equality of two sets
Answer» Correct option: (B) intersection of two set

Discussion

77.	The negation of the statement “ I like Mathematics and English” is (A) I do not like Mathematics and do not like English (B) I like Mathematics but do not like English (C) I do not like Mathematics but like English (D) Either I do not like Mathematics or do not like English
Answer» Correct option: (D) Either I do not like Mathematics or do not like English p : I like Mathematics q : I like English. ~ (p ∧ q ) ≡ ~ p ∨ ~ q ∴ Option (D) is correct.

77.

The negation of the statement “ I like Mathematics and English” is (A) I do not like Mathematics and do not like English (B) I like Mathematics but do not like English (C) I do not like Mathematics but like English (D) Either I do not like Mathematics or do not like English

Answer»

Correct option: (D) Either I do not like Mathematics or do not like English

p : I like Mathematics

q : I like English.

~ (p ∧ q ) ≡ ~ p ∨ ~ q

∴ Option (D) is correct.

Discussion

78.	If the current flows through the given circuit, then it is expressed symbolically as,(A) (p ∧ q) ∨ r (B) (p ∧ q) (C) (p ∨ q) (D) (p ∨ q) ∧ r
Answer» Correct option: (A) (p ∧ q) ∨ r Current will flow in the circuit if switch p and q are closed or switch r is closed. It is represented by (p ∧ q) ∨ r

78.

If the current flows through the given circuit, then it is expressed symbolically as,(A) (p ∧ q) ∨ r (B) (p ∧ q) (C) (p ∨ q) (D) (p ∨ q) ∧ r

Answer»

Correct option: (A) (p ∧ q) ∨ r

Current will flow in the circuit if switch p and q are closed or switch r is closed.

It is represented by

(p ∧ q) ∨ r

Discussion

79.	Which of the following sentences are statements in logic? Justify your answer.i \(\pi\) is a real number.ii. 5! = 120iii. Himalaya is an ocean and Ganga is a river.iv. Please get me a cup of tea.v. Bring me a notebook.vi. Alas ! We lost the matchvii. cos 2\(\theta\) = cos2\(\theta\) -sin2\(\theta\), for all \(\theta\) \(\in\) R.viii. If x is a real number then x2 \(\geq\) 0.
Answer» i. It is a statement. ii. It is a statement. iii. It is a statement. iv. It is an imperative sentence, hence it is not a statement. v. It is an imperative sentence, hence it is not a statement. vi. It is an exclamatory sentence, hence it is not a statement. vii. It is a statement. viii. It is a statement.

79.

Which of the following sentences are statements in logic? Justify your answer.i \(\pi\) is a real number.ii. 5! = 120iii. Himalaya is an ocean and Ganga is a river.iv. Please get me a cup of tea.v. Bring me a notebook.vi. Alas ! We lost the matchvii. cos 2\(\theta\) = cos2\(\theta\) -sin2\(\theta\), for all \(\theta\) \(\in\) R.viii. If x is a real number then x2 \(\geq\) 0.

Answer»

i. It is a statement.

ii. It is a statement.

iii. It is a statement.

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

v. It is an imperative sentence, hence it is not a statement.

vi. It is an exclamatory sentence, hence it is not a statement.

vii. It is a statement.

viii. It is a statement.

Discussion

80.	Write the truth values of the following statements:i. The square of any odd number is even or the cube of any even number is even.ii. \(\sqrt{5}\) is irrational but 3 + \(\sqrt{5}\) is a complex number.iii. \(\exists\) n \(\in\) N, such that n + 5 > 10.
Answer» i. Let p: The square of any odd number is even. q: The cube of any even number is even. \(\therefore\)The symbolic form of the given statement is p \(\lor\) q. Since the truth value of p is F and that of q is T, \(\therefore\) truth value of p \(\lor\) q is T ii. Let p: \(\sqrt{5}\) is irrational. q: 3 +\(\sqrt{5}\) is a complex number. \(\therefore\) The symbolic form of the given statement is p \(\land\) q Since the truth value of p is T and that of q is F, \(\therefore\) truth value of p \(\land\) q is F. iii. Consider the statement, \(\exists\) n \(\in\) N, n + 5 > 10 Clearly n \(\geq\)6, n\(\in\)N satisfy n + 5 > 10. \(\therefore\) its truth value is T.

80.

Write the truth values of the following statements:i. The square of any odd number is even or the cube of any even number is even.ii. \(\sqrt{5}\) is irrational but 3 + \(\sqrt{5}\) is a complex number.iii. \(\exists\) n \(\in\) N, such that n + 5 > 10.

Answer»

i. Let p: The square of any odd number is even.

q: The cube of any even number is even.

\(\therefore\)The symbolic form of the given statement is p \(\lor\) q.

Since the truth value of p is F and that of q is T,

\(\therefore\) truth value of p \(\lor\) q is T

ii. Let p: \(\sqrt{5}\) is irrational.

q: 3 +\(\sqrt{5}\) is a complex number.

\(\therefore\) The symbolic form of the given statement is

p \(\land\) q

Since the truth value of p is T and that of q is F,

\(\therefore\) truth value of p \(\land\) q is F.

iii. Consider the statement, \(\exists\) n \(\in\) N, n + 5 > 10

Clearly n \(\geq\)6, n\(\in\)N satisfy n + 5 > 10.

\(\therefore\) its truth value is T.

Discussion

81.	Which of the following quantified statement is false?(A) ∃x ∈ N, such that x + 5 ≤ 6 (B) ∀x ∈ N, x2 \(\nleq\) 0 (C) ∃x ∈ N, such that x - 1 < 0 (D) ∃x ∈ N, such that x2 - 3x + 2 = 0
Answer» Correct option: (C) ∃x ∈ N, such that x - 1 < 0 Option (C) is false, since for every natural number the statement x - 1 ≥ 0 is always true.

81.

Which of the following quantified statement is false?(A) ∃x ∈ N, such that x + 5 ≤ 6 (B) ∀x ∈ N, x2 \(\nleq\) 0 (C) ∃x ∈ N, such that x - 1 < 0 (D) ∃x ∈ N, such that x2 - 3x + 2 = 0

Answer»

Correct option: (C) ∃x ∈ N, such that x - 1 < 0

Option (C) is false, since for every natural number the statement x - 1 ≥ 0 is always true.

Discussion

82.	Write the truth values of the following statements:i. \(\exists\) n \(\in\) N, n + 3 > 5.ii. If ABC is a triangle and all its sides are equal then each angle has measure 30\(^\circ\).iii. \(\forall\) n \(\in\) N, n2 + n is an even number while n2- n is an odd number.
Answer» i. Consider the statement, \(\forall\) n \(\in\) N, n + 3 > 5 \(\therefore\) n = 1 and n = 2 \(\in\) N do not satisfy n + 3 > 5 \(\therefore\) truth value of p is F. ii. Let p: ABC is a triangle and all its sides are equal. q: Each angle has measure 30\(^\circ\) The symbolic form of the given statement is p \(\rightarrow\) q Since the truth value of p is T and that of q is F, \(\therefore\) truth value of p\(\rightarrow\) q is F iii. Let p: \(\forall\) n \(\in\) N, n² + n is an even number. q: \(\forall\) n \(\in\) N, n² - n is an odd number. \(\therefore\)The symbolic form of the given statement is p \(\land\) q. Since, the truth value of p is T and q is F, \(\therefore\) truth value of p \(\land\)q is F

82.

Write the truth values of the following statements:i. \(\exists\) n \(\in\) N, n + 3 > 5.ii. If ABC is a triangle and all its sides are equal then each angle has measure 30\(^\circ\).iii. \(\forall\) n \(\in\) N, n2 + n is an even number while n2- n is an odd number.

Answer»

i. Consider the statement, \(\forall\) n \(\in\) N, n + 3 > 5

\(\therefore\) n = 1 and n = 2 \(\in\) N do not satisfy n + 3 > 5

\(\therefore\) truth value of p is F.

ii. Let p: ABC is a triangle and all its sides are equal.

q: Each angle has measure 30\(^\circ\)

The symbolic form of the given statement is p \(\rightarrow\) q

Since the truth value of p is T and that of q is F,

\(\therefore\) truth value of p\(\rightarrow\) q is F

iii. Let p: \(\forall\) n \(\in\) N, n² + n is an even number.

q: \(\forall\) n \(\in\) N, n² - n is an odd number.

\(\therefore\)The symbolic form of the given statement is p \(\land\) q.

Since, the truth value of p is T and q is F,

\(\therefore\) truth value of p \(\land\)q is F

Discussion

83.	If A = {4, 5, 7, 9}, determine the truth value of each of the following quantified statements.i. \(\exists\) x \(\in\) A, such that x + 2 = 7.ii. \(\forall\) x \(\in\) A, x + 3 < 10.iii. \(\forall\) x \(\in\) A, such that x + 5 \(\geq\)9.iv. \(\exists\) x \(\in\) A, such that x is even.v. \(\forall\) x \(\in\) A, 2x \(\leq\) 17.
Answer» i. Since x = 5 \(\in\) A, satisfies x + 2 = 7. \(\therefore\)the given statement is true. \(\therefore\) Its truth value is ‘T’. ii. Since, x = 7, 9 \(\in\) A, do not satisfy x + 3 < 10. \(\therefore\) the given statement is false. \(\therefore\) Its truth value is ‘F’. iii. Since, x = 4, 5, 7, 9 \(\in\) A, satisfy x + 5 \(\geq\) 9. \(\therefore\) the given statement is true. \(\therefore\) Its truth value is ‘T’. iv. Since, x = 4 \(\in\) A, satisfies ‘x is even’. \(\therefore\) the given statement is true. \(\therefore\) Its truth value is ‘T’. v. Since x = 9 \(\in\) A does not satisfy 2x \(\leq\) 17. \(\therefore\) the given statement is false. \(\therefore\) Its truth value is ‘F’.

83.

If A = {4, 5, 7, 9}, determine the truth value of each of the following quantified statements.i. \(\exists\) x \(\in\) A, such that x + 2 = 7.ii. \(\forall\) x \(\in\) A, x + 3 < 10.iii. \(\forall\) x \(\in\) A, such that x + 5 \(\geq\)9.iv. \(\exists\) x \(\in\) A, such that x is even.v. \(\forall\) x \(\in\) A, 2x \(\leq\) 17.

Answer»

i. Since x = 5 \(\in\) A, satisfies x + 2 = 7.

\(\therefore\)the given statement is true.

\(\therefore\) Its truth value is ‘T’.

ii. Since, x = 7, 9 \(\in\) A, do not satisfy x + 3 < 10.

\(\therefore\) the given statement is false.

\(\therefore\) Its truth value is ‘F’.

iii. Since, x = 4, 5, 7, 9 \(\in\) A, satisfy x + 5 \(\geq\) 9.

\(\therefore\) the given statement is true.

\(\therefore\) Its truth value is ‘T’.

iv. Since, x = 4 \(\in\) A, satisfies ‘x is even’.

\(\therefore\) the given statement is true.

\(\therefore\) Its truth value is ‘T’.

v. Since x = 9 \(\in\) A does not satisfy 2x \(\leq\) 17.

\(\therefore\) the given statement is false.

\(\therefore\) Its truth value is ‘F’.

Discussion

84.	Write negations of the following statementsi. Some buildings in this area are multistoried.ii. All parents care for their children.iii. \(\forall\) n \(\in\) N, n + 7 > 6.iv. \(\exists\) x \(\in\) A, such that x + 5 > 8.
Answer» i. All buildings in this area are not multistoried. ii. Some parents do not care for their children. iii. \(\exists\) n \(\in\) N, such that n + 7 \(\leq\) 6. iv. \(\forall\) x\(\in\) A, x + 5 \(\leq\) 8.

84.

Write negations of the following statementsi. Some buildings in this area are multistoried.ii. All parents care for their children.iii. \(\forall\) n \(\in\) N, n + 7 > 6.iv. \(\exists\) x \(\in\) A, such that x + 5 > 8.

Answer»

i. All buildings in this area are not multistoried.

ii. Some parents do not care for their children.

iii. \(\exists\) n \(\in\) N, such that n + 7 \(\leq\) 6.

iv. \(\forall\) x\(\in\) A, x + 5 \(\leq\) 8.

Discussion

85.	Write the following statements in symbolic form:i. Ramesh is cruel or strict.ii. I am brave is necessary and sufficient condition to climb the Mount Everest.iii. I can travel by train provided I get my ticket reserved.iv. Sandeep neither likes tea nor coffee but enjoys a soft-drink.v. ABC is a triangle only if AB + BC > AC.vi. Rajesh is studious but does not get good marks.
Answer» i. Let p: Ramesh is cruel, q: Ramesh is strict. \(\therefore\)The symbolic form of the given statement is p \(\lor\) q. ii. Let p: I am brave. q: I can climb the Mount Everest. \(\therefore\) The symbolic form of the given statement is p \(\leftrightarrow\) q. iii. Let p: I can travel by train, q: I get my ticket reserved. \(\therefore\)The symbolic form of the given statement is q \(\leftrightarrow\) p. iv. Let p: Sandeep likes tea, q: Sandeep likes coffee. r: Sandeep enjoys a soft-drink. \(\therefore\) The symbolic form of the given statement is (~p \(\land\) ~q) \(\land\) r. v. Let p: ABC is a triangle, q: AB + BC > AC. \(\therefore\) The symbolic form of the given statement is p \(\rightarrow\) q. vi. Let p: Rajesh is studious, q: Rajesh gets good marks. \(\therefore\) The symbolic form of the given statement is p \(\land\) ~q.

85.

Write the following statements in symbolic form:i. Ramesh is cruel or strict.ii. I am brave is necessary and sufficient condition to climb the Mount Everest.iii. I can travel by train provided I get my ticket reserved.iv. Sandeep neither likes tea nor coffee but enjoys a soft-drink.v. ABC is a triangle only if AB + BC > AC.vi. Rajesh is studious but does not get good marks.

Answer»

i. Let p: Ramesh is cruel,

q: Ramesh is strict.

\(\therefore\)The symbolic form of the given statement is p \(\lor\) q.

ii. Let p: I am brave.

q: I can climb the Mount Everest.

\(\therefore\) The symbolic form of the given statement is p \(\leftrightarrow\) q.

iii. Let p: I can travel by train,

q: I get my ticket reserved.

\(\therefore\)The symbolic form of the given statement is q \(\leftrightarrow\) p.

iv. Let p: Sandeep likes tea,

q: Sandeep likes coffee.

r: Sandeep enjoys a soft-drink.

\(\therefore\) The symbolic form of the given statement is

(~p \(\land\) ~q) \(\land\) r.

v. Let p: ABC is a triangle,

q: AB + BC > AC.

\(\therefore\) The symbolic form of the given statement is p \(\rightarrow\) q.

vi. Let p: Rajesh is studious,

q: Rajesh gets good marks.

\(\therefore\) The symbolic form of the given statement is p \(\land\) ~q.

Discussion

86.	Without using truth table prove that :(p ∧ q) ∨ (~ p ∧ q) ∨ (p ∧ ~q) ≡ p ∨ q
Answer» LHS = (p ∧ q) v (~p ∧ q) ∨ (p ∧ ~q) ≡ [(p ∨ ~p) ∧ q] ∨ (p ∧ ~q) … (Distributive Law) ≡ (T ∧ q) ∨ (p ∧ ~q) … (Complement Law) ≡ q ∨ (p ∧ ~q) … (Identity Law) ≡ (q ∨ p) ∧ (q ∨ ~q) … (Distributive Law) ≡ (q ∨ p) ∧ T .. (Complement Law) ≡ q ∨ p … (Identity Law) ≡ p ∨ q … (Commutative Law) ≡ RHS.

86.

Without using truth table prove that :(p ∧ q) ∨ (~ p ∧ q) ∨ (p ∧ ~q) ≡ p ∨ q

Answer»

LHS = (p ∧ q) v (~p ∧ q) ∨ (p ∧ ~q)

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

≡ (T ∧ q) ∨ (p ∧ ~q) … (Complement Law)

≡ q ∨ (p ∧ ~q) … (Identity Law)

≡ (q ∨ p) ∧ (q ∨ ~q) … (Distributive Law)

≡ (q ∨ p) ∧ T .. (Complement Law)

≡ q ∨ p … (Identity Law)

≡ p ∨ q … (Commutative Law)

≡ RHS.

Discussion

87.	Without using truth table prove that :(p ∨ q) ∧ (p ∨ ~q) ≡ p
Answer» LHS = (p ∨ q) ∧ (p ∨ ~q) ≡ p ∨ (q ∧ ~q) … (Distributive Law) ≡ p ∨ F … (Complement Law) ≡ p … (Identity Law) ≡ RHS.

87.

Without using truth table prove that :(p ∨ q) ∧ (p ∨ ~q) ≡ p

Answer»

LHS = (p ∨ q) ∧ (p ∨ ~q)

≡ p ∨ (q ∧ ~q) … (Distributive Law)

≡ p ∨ F … (Complement Law)

≡ p … (Identity Law)

≡ RHS.

Discussion

88.	The proposition p → ~ (p ∧ q) is a(A) tautology(B) contradiction(C) contingency (D) either (A) or (B)
Answer» Correct answer: (C) contingency

Discussion

89.	The false statement in the following is(A) p \(\lor\) (~p) is a contradiction.(B) (p \(\rightarrow\) q) \(\rightarrow\) (~q \(\rightarrow\) ~p) is a contradiction.(C) ~(~p)\(\rightarrow\) p is a tautology.(D) p \(\lor\) (~p) is a tautology.
Answer» Correct answer: (B) (p \(\rightarrow\) q) \(\rightarrow\) (~q \(\rightarrow\) ~p) is a contradiction.

Discussion

90.	If m: Rimi likes calculus. n: Rimi opts for engineering branch. Then the verbal form of m → n is (A) If Rimi opts for engineering branch then she likes calculus. (B) If Rimi likes calculus then she does not opt for engineering branch. (C) If Rimi likes calculus then she opts for engineering branch (D) If Rimi likes engineering branch then she opts for calculus.
Answer» Correct option: (C) If Rimi likes calculus then she opts for engineering branch ‘m → n’ means ‘If m then n’, ∴ option (C) is correct.

90.

If m: Rimi likes calculus. n: Rimi opts for engineering branch. Then the verbal form of m → n is (A) If Rimi opts for engineering branch then she likes calculus. (B) If Rimi likes calculus then she does not opt for engineering branch. (C) If Rimi likes calculus then she opts for engineering branch (D) If Rimi likes engineering branch then she opts for calculus.

Answer»

Correct option: (C) If Rimi likes calculus then she opts for engineering branch

‘m → n’ means ‘If m then n’,

∴ option (C) is correct.

Discussion

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

91.

Write the duals of p ∨ (q ∧ r).

Answer»

The duals of the given statement patterns are :

p ∧ (q ∨ r)

Discussion

92.	If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following :(i) Ǝ x ∈ A such that x – 8 = 1(ii) \(\forall\) x ∈ A, x2 + x is an even number.(iii) Ǝ x ∈ A such that x2 < 0(iv) \(\forall\) x ∈ A, x is an even number.(v) Ǝ x ∈ A such that 3x + 8 > 40(vi) \(\forall\) x ∈ A, 2x + 9 > 14
Answer» (i) Ǝ x ∈ A such that x – 8 = 1 Clearly x = 9 ∈ A satisfies x – 8 = 1. So the given statement is true, hence its truth value is T. (ii) \(\forall\) x ∈ A, x² + x is an even number. For each x ∈ A, x² + x is an even number. So the given statement is true, hence its truth value is T. (iii) Ǝ x ∈ A such that x² < 0 There is no x ∈ A which satisfies x² < 0. So the given statement is false, hence its truth value is F. (iv) \(\forall\) x ∈ A, x is an even number. x = 3 ∈ A, x = 5 ∈ A, x = 7 ∈ A, x = 9 ∈ A, x = 11 ∈ A do not satisfy x is an even number. So the given statement is false, hence its truth value is F. (v) Ǝ x ∈ A such that 3x + 8 > 40 Clearly x = 11 ∈ A and x = 12 ∈ A satisfies 3x + 8 > 40. So the given statement is true, hence its truth value is T. (vi) \(\forall\) x ∈ A, 2x + 9 > 14 For each x ∈ A, 2x + 9 > 14. So the given statement is true, hence its truth value is T.

92.

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following :(i) Ǝ x ∈ A such that x – 8 = 1(ii) \(\forall\) x ∈ A, x2 + x is an even number.(iii) Ǝ x ∈ A such that x2 < 0(iv) \(\forall\) x ∈ A, x is an even number.(v) Ǝ x ∈ A such that 3x + 8 > 40(vi) \(\forall\) x ∈ A, 2x + 9 > 14

Answer»

(i) Ǝ x ∈ A such that x – 8 = 1

Clearly x = 9 ∈ A satisfies x – 8 = 1.

So the given statement is true, hence its truth value is T.

(ii) \(\forall\) x ∈ A, x² + x is an even number.

For each x ∈ A,

x² + x is an even number.

So the given statement is true, hence its truth value is T.

(iii) Ǝ x ∈ A such that x² < 0

There is no x ∈ A which satisfies x² < 0.

So the given statement is false, hence its truth value is F.

(iv) \(\forall\) x ∈ A, x is an even number.

x = 3 ∈ A, x = 5 ∈ A, x = 7 ∈ A, x = 9 ∈ A, x = 11 ∈ A do not satisfy x is an even number.

So the given statement is false, hence its truth value is F.

(v) Ǝ x ∈ A such that 3x + 8 > 40

Clearly x = 11 ∈ A and x = 12 ∈ A satisfies 3x + 8 > 40.

So the given statement is true, hence its truth value is T.

(vi) \(\forall\) x ∈ A, 2x + 9 > 14

For each x ∈ A,

2x + 9 > 14.

So the given statement is true, hence its truth value is T.

Discussion

93.	Write the truth values of the following statements :Ǝ n ∈ N such that n + 5 > 10.
Answer» Ǝ n ∈ N, such that n + 5 > 10 is a true statement, hence its truth value is T. (All n ≥ 6, where n ∈ N, satisfy n + 5 > 10).

93.

Write the truth values of the following statements :Ǝ n ∈ N such that n + 5 > 10.

Answer»

Ǝ n ∈ N, such that n + 5 > 10 is a true statement, hence its truth value is T.

(All n ≥ 6, where n ∈ N, satisfy n + 5 > 10).

Discussion

94.	Write the truth values of the following statements :\(\forall\) n ∈ N, n + 6 > 8.
Answer» \(\forall\) n ∈ N, n + 6 > 8 is a false statement, hence its truth value is F. {n = 1 ∈ N, n = 2 ∈ N do not satisfy n + 6 > 8).

94.

Write the truth values of the following statements :\(\forall\) n ∈ N, n + 6 > 8.

Answer»

\(\forall\) n ∈ N, n + 6 > 8 is a false statement, hence its truth value is F.

{n = 1 ∈ N, n = 2 ∈ N do not satisfy n + 6 > 8).

Discussion

95.	Write the truth values of the following statements :In ∆ ABC if all sides are equal then its all angles are equal.
Answer» Let p : ABC is a triangle and all its sides are equal. q : Its all angles are equal. Then, The symbolic form of the given statement is p → q If the truth value of p is T, Then, The truth value of q is T. ∴ The truth value of p → q is T. … [T → T ≡ T].

95.

Write the truth values of the following statements :In ∆ ABC if all sides are equal then its all angles are equal.

Answer»

Let p : ABC is a triangle and all its sides are equal.

q : Its all angles are equal.

Then,

The symbolic form of the given statement is p → q

If the truth value of p is T,

Then,

The truth value of q is T.

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

Discussion

96.	Write the negations of the following :(i) Tirupati is in Andhra Pradesh.(ii) 3 is not a root of the equation x2 + 3x – 18 = 0.(iii) \(\sqrt2\) is a rational number.(iv) Polygon ABCDE is a pentagon.(v) 7 + 3 > 5.
Answer» (i) The negations of the given statements are : Tirupati is not in Andhra Pradesh. (ii) 3 is a root of the equation x² + 3x – 18 = 0. (iii) \(\sqrt2\) is a rational number. (iv) Polygon ABCDE is not a pentagon. (v) 7 + 3 > 5.

96.

Write the negations of the following :(i) Tirupati is in Andhra Pradesh.(ii) 3 is not a root of the equation x2 + 3x – 18 = 0.(iii) \(\sqrt2\) is a rational number.(iv) Polygon ABCDE is a pentagon.(v) 7 + 3 > 5.

Answer»

(i) The negations of the given statements are :

Tirupati is not in Andhra Pradesh.

(ii) 3 is a root of the equation x² + 3x – 18 = 0.

(iii) \(\sqrt2\) is a rational number.

(iv) Polygon ABCDE is not a pentagon.

(v) 7 + 3 > 5.

Discussion

97.	Write converse, inverse and contrapositive of :If surface area decreases then pressure increases.
Answer» Let p : The surface area decreases. q : The pressure increases. Then, The symbolic form of the given statement is p → q. Converse : q → p is the converse of p→ q. i.e. If the pressure increases, then the surface area decreases. Contrapositive : ~q → ~p is the contrapositive of p → q. i.e. If the pressure does not increase, then the surface area does not decrease.

97.

Write converse, inverse and contrapositive of :If surface area decreases then pressure increases.

Answer»

Let p : The surface area decreases.

q : The pressure increases.

Then,

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

Converse : q → p is the converse of p→ q.

i.e. If the pressure increases, then the surface area decreases.

Contrapositive : ~q → ~p is the contrapositive of p → q.

i.e. If the pressure does not increase, then the surface area does not decrease.

Discussion

98.	Write converse, inverse and contrapositive of :If x < y then x2 < y2 (x, y ∈ R)
Answer» Let p : x < y, q : x < y . Then, The symbolic form of the given statement is p → q. Converse : q → p is the converse of p → q. i.e. If x² < y², then x < y. Inverse : ~p → ~q is the inverse of p → q. i.e. If x ≯ y, then x²≯ y². OR If x ≮ y, then x²≮ y². Contrapositive : ~q → p is the contrapositive of p → q i.e. If x² ≯ y², then x ≯ y. OR If x ≮ y , then x ≮ y.

98.

Write converse, inverse and contrapositive of :If x < y then x2 < y2 (x, y ∈ R)

Answer»

Let p : x < y,

q : x < y .

Then,

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

Converse : q → p is the converse of p → q.

i.e. If x² < y², then x < y.

Inverse : ~p → ~q is the inverse of p → q.

i.e. If x ≯ y, then x²≯ y².

OR

If x ≮ y, then x²≮ y².

Contrapositive : ~q → p is the contrapositive of p → q

i.e. If x² ≯ y², then x ≯ y.

OR

If x ≮ y , then x ≮ y.

Discussion

99.	Write converse, inverse and contrapositive of :A family becomes literate if the woman in it is literate.
Answer» Let p : The woman in the family is literate. q : A family become literate. Then, The symbolic form of the given statement is p → q Converse : q → p is the converse of p → q. i.e. If a family become literate, then the woman in it is literate. Inverse : ~p → ~q is the inverse of p → q. i.e. If the woman in the family is not literate, then the family does not become literate. Contrapositive : ~q → ~p is the contrapositive of p → q. i e. If a family does not become literate, then the woman in it is not literate.

99.

Write converse, inverse and contrapositive of :A family becomes literate if the woman in it is literate.

Answer»

Let p : The woman in the family is literate.

q : A family become literate.

Then,

The symbolic form of the given statement is p → q

Converse : q → p is the converse of p → q.

i.e. If a family become literate, then the woman in it is literate.

Inverse : ~p → ~q is the inverse of p → q.

i.e. If the woman in the family is not literate,

then the family does not become literate.

Contrapositive : ~q → ~p is the contrapositive of p → q.

i e. If a family does not become literate, then the woman in it is not literate.

Discussion

100.

The converse of the contrapositive of p → q is(A) ~p → q (B) p → ~q (C) ~p → ~q (D) ~q → p

Answer»

Correct option: (C) ~p → ~q

Given p → q

Its contrapositive is ~q → ~p

and its converse is ~p → ~q

Discussion

Explore topic-wise InterviewSolutions in .

A compound statement p or q is false only when (A) p is false. (B) q is false. (C) both p and q are false. (D) depends on p and q.

Assuming p: She is beautiful,q: She is clever, the verbal form of p\(\land\) (~q) is(A) She is beautiful but not clever.(B) She is beautiful and clever.(C) She is not beautiful and not clever.(D) She is beautiful or not clever.

Using the statementsp: Kiran passed the examination,s : Kiran is sad.the statement ‘It is not true that Kiran passestherefore he is sad’ in symbolic form is(A) ~p \(\rightarrow\) s (B) ~ (p \(\rightarrow\) ~ s) (C) ~p \(\rightarrow\) ~ s (D) ~ (p \(\rightarrow\) s)

Assuming the first part of each statement as p, second as q and the third as r, the statement ‘If A, B, C are three distinct points, then either they are collinear or they form a triangle’ in symbolic form is (A) p ↔ (q ∨ r) (B) (p ∧ q) → r (C) p → (q ∨ r) (D) p → (q ∧ r)

Assuming p: She is beautiful, q: She is clever,the verbal form of ~p \(\land\) (~q) is(A) She is beautiful but not clever.(B) She is beautiful and clever.(C) She is not beautiful and not clever.(D) She is beautiful or not clever.

If A = {2, 3, 4, 5, 6}, then which of the following is not true?(A) \(\exists\) x \(\in\) A such that x + 3 = 8(B) \(\exists\) x \(\in\) A such that x + 2 &lt; 5(C) \(\exists\) x \(\in\) A such that x + 2 &lt; 9(D) \(\forall\) x \(\in\) A such that x + 6 \(\geq\) 9

If p and q be two statements then the conjunction of the statements, p \(\land\) q is false when(A) both p and q are true. (B) either p or q are true(C) either p or q or both are false. (D) both p and q are false.

When two statements are connected by the connective ‘if and only if’ then the compound statement is called(A) conjunction of the statements.(B) disjunction of the statements.(C) biconditional statement.(D) conditional statement.

Negation of ~(p \(\lor\) q) is(A) ~p \(\lor\) ~q(B) ~p \(\land\) ~q(C) p \(\land\) ~q(D) p \(\lor\) ~q

The dual of ~(p\(\lor\)q) \(\lor\)[p\(\lor\)(q\(\land\)~r)] is, (A) ~(p\(\land\)q) \(\land\) [p \(\lor\) (q\(\land\)~r)](B) (p\(\land\)q) \(\land\) [p \(\land\) (q \(\lor\) ~ r)](C) ~(p\(\land\)q) \(\land\) [p \(\land\) (q\(\land\) r)](D) ~(p\(\land\)q) \(\land\) [p \(\land\) (q \(\lor\) ~ r)]

Using rules in logic, prove the following : (i) p ↔ q ≡ ~ (p ∧ ~q) ∧ ~(q ∧ ~p)(ii) ~p ∧ q ≡ (p ∨ q) ∧ ~p(iii) ~(p ∨ q) ∨ (~p ∧ q) ≡ ~p

Given that p is ‘false’ and q is ‘true’ then the statement which is ‘false’ is(A) ~p → ~q(B) p → (q ∧ p) (C) p → ~q (D) q → ~p

If p, q are true and r is false statement then which of the following is true statement? (A) (p ∧ q) ∨ r is F (B) (p ∧ q) → r is T (C) (p ∨ q) ∧ (p ∨ r) is T (D) (p → q) ↔ (p → r) is T

f ~q ∨ p is F, then which of the following is correct?(A) p ↔ q is T (B) p → q is T (C) q → p is T (D) p → q is F

~ (p ∨ q) ∨ (~p ∧ q) is logically equivalent to (A) ~p (B) p (C) q (D) ~q

p ∧ (p → q) is logically equivalent to (A) p ∨ q (B) ~p ∨ q (C) p ∧ q (D) p ∨ ~q

Negation of a statement in logic corresponds to ........... in set theory. (A) empty set (B) null set (C) complement of a set (D) universal set

If p and q are two logical statements and A and B are two sets, then p → q corresponds to (A) A ⊆ B (B) A ∩ B (C) A ∪ B (D) A \(\nsubseteq\) B

The logical statement ‘p ∧ q’ can be related to the set theory’s concept of (A) union of two sets (B) intersection of two set (C) subset of a set (D) equality of two sets

The negation of the statement “ I like Mathematics and English” is (A) I do not like Mathematics and do not like English (B) I like Mathematics but do not like English (C) I do not like Mathematics but like English (D) Either I do not like Mathematics or do not like English

If the current flows through the given circuit, then it is expressed symbolically as,(A) (p ∧ q) ∨ r (B) (p ∧ q) (C) (p ∨ q) (D) (p ∨ q) ∧ r

Write the truth values of the following statements:i. The square of any odd number is even or the cube of any even number is even.ii. \(\sqrt{5}\) is irrational but 3 + \(\sqrt{5}\) is a complex number.iii. \(\exists\) n \(\in\) N, such that n + 5 &gt; 10.

Which of the following quantified statement is false?(A) ∃x ∈ N, such that x + 5 ≤ 6 (B) ∀x ∈ N, x2 \(\nleq\) 0 (C) ∃x ∈ N, such that x - 1 &lt; 0 (D) ∃x ∈ N, such that x2 - 3x + 2 = 0

Write the truth values of the following statements:i. \(\exists\) n \(\in\) N, n + 3 &gt; 5.ii. If ABC is a triangle and all its sides are equal then each angle has measure 30\(^\circ\).iii. \(\forall\) n \(\in\) N, n2 + n is an even number while n2- n is an odd number.

Write negations of the following statementsi. Some buildings in this area are multistoried.ii. All parents care for their children.iii. \(\forall\) n \(\in\) N, n + 7 &gt; 6.iv. \(\exists\) x \(\in\) A, such that x + 5 &gt; 8.

Without using truth table prove that :(p ∧ q) ∨ (~ p ∧ q) ∨ (p ∧ ~q) ≡ p ∨ q

Without using truth table prove that :(p ∨ q) ∧ (p ∨ ~q) ≡ p

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

The false statement in the following is(A) p \(\lor\) (~p) is a contradiction.(B) (p \(\rightarrow\) q) \(\rightarrow\) (~q \(\rightarrow\) ~p) is a contradiction.(C) ~(~p)\(\rightarrow\) p is a tautology.(D) p \(\lor\) (~p) is a tautology.

Write the duals of p ∨ (q ∧ r).

Write the truth values of the following statements :Ǝ n ∈ N such that n + 5 &gt; 10.

Write the truth values of the following statements :\(\forall\) n ∈ N, n + 6 &gt; 8.

Write the truth values of the following statements :In ∆ ABC if all sides are equal then its all angles are equal.

Write the negations of the following :(i) Tirupati is in Andhra Pradesh.(ii) 3 is not a root of the equation x2 + 3x – 18 = 0.(iii) \(\sqrt2\) is a rational number.(iv) Polygon ABCDE is a pentagon.(v) 7 + 3 &gt; 5.

Write converse, inverse and contrapositive of :If surface area decreases then pressure increases.

Write converse, inverse and contrapositive of :If x &lt; y then x2 &lt; y2 (x, y ∈ R)

Write converse, inverse and contrapositive of :A family becomes literate if the woman in it is literate.

The converse of the contrapositive of p → q is(A) ~p → q (B) p → ~q (C) ~p → ~q (D) ~q → p

If A = {2, 3, 4, 5, 6}, then which of the following is not true?(A) \(\exists\) x \(\in\) A such that x + 3 = 8(B) \(\exists\) x \(\in\) A such that x + 2 < 5(C) \(\exists\) x \(\in\) A such that x + 2 < 9(D) \(\forall\) x \(\in\) A such that x + 6 \(\geq\) 9

Write the truth values of the following statements:i. The square of any odd number is even or the cube of any even number is even.ii. \(\sqrt{5}\) is irrational but 3 + \(\sqrt{5}\) is a complex number.iii. \(\exists\) n \(\in\) N, such that n + 5 > 10.

Which of the following quantified statement is false?(A) ∃x ∈ N, such that x + 5 ≤ 6 (B) ∀x ∈ N, x2 \(\nleq\) 0 (C) ∃x ∈ N, such that x - 1 < 0 (D) ∃x ∈ N, such that x2 - 3x + 2 = 0

Write the truth values of the following statements:i. \(\exists\) n \(\in\) N, n + 3 > 5.ii. If ABC is a triangle and all its sides are equal then each angle has measure 30\(^\circ\).iii. \(\forall\) n \(\in\) N, n2 + n is an even number while n2- n is an odd number.

Write negations of the following statementsi. Some buildings in this area are multistoried.ii. All parents care for their children.iii. \(\forall\) n \(\in\) N, n + 7 > 6.iv. \(\exists\) x \(\in\) A, such that x + 5 > 8.

Write the truth values of the following statements :Ǝ n ∈ N such that n + 5 > 10.

Write the truth values of the following statements :\(\forall\) n ∈ N, n + 6 > 8.

Write the negations of the following :(i) Tirupati is in Andhra Pradesh.(ii) 3 is not a root of the equation x2 + 3x – 18 = 0.(iii) \(\sqrt2\) is a rational number.(iv) Polygon ABCDE is a pentagon.(v) 7 + 3 > 5.

Write converse, inverse and contrapositive of :If x < y then x2 < y2 (x, y ∈ R)