Class 12 Mathematics 1 Mathematical Logic Exercise 1.1 Solutions | Maharashtra Board

 

Class 12 Mathematics 1

Mathematical Logic

Exercise 1.1

---

Q.1. State which of the following are statements. Justify. In case of statement, state its truth value.

i) 5 + 4 = 13.

It is a statement.

Truth value: False (F)

ii) x − 3 = 14.

It is not a statement because the value of x is not specified.

It is an open sentence.

iii) Close the door.

It is not a statement because it is a command.

iv) Zero is a complex number.

It is a statement.

Truth value: True (T)

v) Please get me breakfast.

It is not a statement because it is a request.

vi) Congruent triangles are similar.

It is a statement.

Truth value: True (T)

vii) x² = x.

It is not a statement because the value of x is not specified.

viii) A quadratic equation cannot have more than two roots.

It is a statement.

Truth value: True (T)

ix) Do you like Mathematics?

It is not a statement because it is a question.

x) The sun sets in the west.

It is a statement.

Truth value: True (T)

xi) All real numbers are whole numbers.

It is a statement.

Truth value: False (F)

xii) Can you speak in Marathi?

It is not a statement because it is a question.

xiii) x² − 6x − 7 = 0, when x = 7.

Substituting x = 7:

49 − 42 − 7 = 0

0 = 0

It is a statement.

Truth value: True (T)

xiv) The sum of cuberoots of unity is zero.

It is a statement.

Truth value: True (T)

xv) It rains heavily.

It is a statement because it can be either true or false depending on the place and time.

Its truth value cannot be determined without further information.

---

Q.2. Write the following compound statements symbolically.

i) Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Let p: Nagpur is in Maharashtra.

Let q: Chennai is in Tamil Nadu.

Answer: p ∧ q

ii) Triangle is equilateral or isosceles.

Let p: Triangle is equilateral.

Let q: Triangle is isosceles.

Answer: p ∨ q

iii) The angle is right angle if and only if it is of measure 90°.

Let p: The angle is a right angle.

Let q: The angle measures 90°.

Answer: p ↔ q

iv) Angle is neither acute nor obtuse.

Let p: The angle is acute.

Let q: The angle is obtuse.

Answer: ¬p ∧ ¬q

v) If ΔABC is right angled at B, then m∠A + m∠C = 90°.

Let p: ΔABC is right angled at B.

Let q: m∠A + m∠C = 90°.

Answer: p → q

vi) Hima Das wins gold medal if and only if she runs fast.

Let p: Hima Das wins gold medal.

Let q: Hima Das runs fast.

Answer: p ↔ q

vii) x is not irrational number but is a square of an integer.

Let p: x is an irrational number.

Let q: x is a square of an integer.

Answer: ¬p ∧ q

---

Q.3. Write the truth values of the following.

No.	Statement	Truth Value
i)	4 is odd or 1 is prime.	True (T)
ii)	64 is a perfect square and 46 is a prime number.	False (F)
iii)	5 is a prime number and 7 divides 94.	False (F)
iv)	It is not true that 5 − 3i is a real number.	True (T)
v)	If 3 × 5 = 8 then 3 + 5 = 15.	True (T)
vi)	Milk is white if and only if sky is blue.	False (F)
vii)	24 is a composite number or 17 is a prime number.	True (T)

---

Q.4. If the statements p, q are true statements and r, s are false statements then determine the truth values.

Given:

p = T, q = T, r = F, s = F

No.	Statement	Truth Value
i)	p ∨ (q ∧ r)	T
ii)	(p → q) ∨ (r → s)	T
iii)	(q ∧ r) ∨ (¬p ∧ s)	F
iv)	(p → q) ∧ ¬r	T
v)	(¬r ↔ p) → ¬q	F
vi)	[¬p ∧ (¬q ∧ r)] ∨ [(q ∧ r) ∨ (p ∧ r)]	F
vii)	[(¬p ∧ q) ∧ ¬r] ∨ [(q → p) → (¬s ∨ r)]	T
viii)	¬[(¬p ∧ r) ∨ (s → ¬q)] ↔ (p ∧ r)	T

---

Q.5. Write the negations of the following.

i) Tirupati is in Andhra Pradesh.

Negation: Tirupati is not in Andhra Pradesh.

ii) 3 is not a root of the equation x² + 3x − 18 = 0.

Negation: 3 is a root of the equation x² + 3x − 18 = 0.

iii) √2 is a rational number.

Negation: √2 is an irrational number.

iv) Polygon ABCDE is a pentagon.

Negation: Polygon ABCDE is not a pentagon.

v) 7 + 3 > 5

Negation: 7 + 3 ≤ 5

---

Rukmini Classes | Class 11 Mathematics