Class 12 Mathematics 1 Mathematical Logic Practice Set 1.3 Solutions Labels

 

Class 12 Mathematics 1
Mathematical Logic – Practice Set 1.3

Question 1

Q.1. If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following.

i) ∃ x ∈ A such that x − 8 = 1
ii) ∀ x ∈ A, x² + x is an even number
iii) ∃ x ∈ A such that x² < 0
iv) ∀ x ∈ A, x is an even number
v) ∃ x ∈ A such that 3x + 8 > 40
vi) ∀ x ∈ A, 2x + 9 > 14

Solution:

i) x − 8 = 1
x = 9 and 9 ∈ A.
Therefore, the statement is True.

ii) x² + x = x(x + 1)
The product of two consecutive integers is always even.
Therefore, the statement is True.

iii) For every real number x, x² ≥ 0.
Therefore, x² < 0 is impossible.
Hence, the statement is False.

iv) All elements of A are not even. For example, 3 is odd.
Therefore, the statement is False.

v) 3x + 8 > 40
3x > 32
x > 32/3
For x = 11 or 12, the inequality is satisfied.
Therefore, the statement is True.

vi) 2x + 9 > 14
2x > 5
x > 2.5
Every element of A is greater than 2.5.
Therefore, the statement is True.

Question 2 (i–v)

Q.2. Write the duals of each of the following.

i) p ∨ (q ∧ r)
ii) p ∧ (q ∧ r)
iii) (p ∨ q) ∧ (r ∨ s)
iv) p ∧ ~q
v) (~p ∨ q) ∧ (~r ∧ s)

Rule: To obtain the dual, interchange ∨ and ∧.

i) p ∨ (q ∧ r)
Dual = p ∧ (q ∨ r)

ii) p ∧ (q ∧ r)
Dual = p ∨ (q ∨ r)

iii) (p ∨ q) ∧ (r ∨ s)
Dual = (p ∧ q) ∨ (r ∧ s)

iv) p ∧ ~q
Dual = p ∨ ~q

v) (~p ∨ q) ∧ (~r ∧ s)
Dual = (~p ∧ q) ∨ (~r ∨ s)

Question 2 (vi–x)

Q.2. Write the duals of each of the following.

vi) ~p ∧ (~q ∧ (p ∨ q) ∧ ~r)
vii) [~(p ∨ q)] ∧ [p ∨ ~(q ∧ ~s)]
viii) c ∨ {p ∧ (q ∨ r)}
ix) ~p ∨ (q ∧ r) ∧ t
x) (p ∨ q) ∨ c

Rule: Interchange ∨ and ∧.

vi) ~p ∧ (~q ∧ (p ∨ q) ∧ ~r)
Dual = ~p ∨ (~q ∨ (p ∧ q) ∨ ~r)

vii) [~(p ∨ q)] ∧ [p ∨ ~(q ∧ ~s)]
Dual = [~(p ∧ q)] ∨ [p ∧ ~(q ∨ ~s)]

viii) c ∨ {p ∧ (q ∨ r)}
Dual = c ∧ {p ∨ (q ∧ r)}

ix) ~p ∨ (q ∧ r) ∧ t
Dual = ~p ∧ (q ∨ r) ∨ t

x) (p ∨ q) ∨ c
Dual = (p ∧ q) ∧ c

Question 3 (i–v)

Q.3. Write the negations of the following.

i) x + 8 > 11 or y − 3 = 6
ii) 11 < 15 and 25 > 20
iii) Quadrilateral is a square if and only if it is a rhombus.
iv) It is cold and raining.
v) If it is raining then we will go and play football.

Rules:
Negation of p ∨ q = ~p ∧ ~q
Negation of p ∧ q = ~p ∨ ~q
Negation of p → q = p ∧ ~q
Negation of p ↔ q = p ⊻ q

i) Original statement: x + 8 > 11 or y − 3 = 6
Negation: x + 8 ≤ 11 and y − 3 ≠ 6

ii) Original statement: 11 < 15 and 25 > 20
Negation: 11 ≥ 15 or 25 ≤ 20

iii) Original statement: A quadrilateral is a square if and only if it is a rhombus.
Negation: A quadrilateral is a square but it is not a rhombus, or it is a rhombus but it is not a square.

iv) Original statement: It is cold and raining.
Negation: It is not cold or it is not raining.

v) Original statement: If it is raining then we will go and play football.
Negation: It is raining and we will not go and play football.

Question 3 (vi–ix)

Q.3. Write the negations of the following.

vi) 2 is a rational number.
vii) All natural numbers are whole numbers.
viii) ∀ n ∈ N, n² + n + 2 is divisible by 4.
ix) ∃ x ∈ N such that x − 17 < 20

vi) Original statement: 2 is a rational number.
Negation: 2 is not a rational number.

vii) Original statement: All natural numbers are whole numbers.
Negation: There exists at least one natural number which is not a whole number.

viii) Original statement: ∀ n ∈ N, n² + n + 2 is divisible by 4.
Negation: ∃ n ∈ N such that n² + n + 2 is not divisible by 4.

ix) Original statement: ∃ x ∈ N such that x − 17 < 20
Negation: ∀ x ∈ N, x − 17 ≥ 20.