### Chapter 1: Mathematical Logic HSC

Chapter 1: Mathematical Logic

Exercise 1.1

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board

#### EXERCISE 1.1Q 1    PAGE 2

Exercise 1.1 | Q 1 | Page 2

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## A triangle has ‘n’ sides

### SOLUTION

It is an open sentence. Hence, it is not a statement.

[Note: Answer given in the textbook is ‘it is a statement’. However, we found that ‘It is not a statement’.]

#### EXERCISE 1.1Q 2    PAGE 2

Exercise 1.1 | Q 2 | Page 2

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## The sum of interior angles of a triangle is 180°

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 3    PAGE 2

Exercise 1.1 | Q 3 | Page 2

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## You are amazing!

### SOLUTION

It is an exclamatory sentence. Hence, it is not a statement.

#### EXERCISE 1.1Q 4    PAGE 2

Exercise 1.1 | Q 4 | Page 2

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## Please grant me a loan.

### SOLUTION

It is a request. Hence, it is not a statement.

#### EXERCISE 1.1Q 5    PAGE 2

Exercise 1.1 | Q 5 | Page 2

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## √-4  is an irrational number.

### SOLUTION

It is a statement which is false. Hence, it’s truth value is F.

#### EXERCISE 1.1Q 6   PAGE 2

Exercise 1.1 | Q 6 | Page 2

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## x2 − 6x + 8 = 0 implies x = −4 or x = −2.

### SOLUTION

It is a statement which is false. Hence, it’s truth value if F.

#### EXERCISE 1.1Q 7    PAGE 3

Exercise 1.1 | Q 7 | Page 3

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## He is an actor.

### SOLUTION

It is an open sentence. Hence, it is not a statement.

#### EXERCISE 1.1Q 8    PAGE 3

Exercise 1.1 | Q 8 | Page 3

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## Did you eat lunch yet?

### SOLUTION

It is an interrogative sentence. Hence, it is not a statement.

#### EXERCISE 1.1Q 9   PAGE 3

Exercise 1.1 | Q 9 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## Have a cup of cappuccino.

### SOLUTION

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

#### EXERCISE 1.1Q 10    PAGE 3

Exercise 1.1 | Q 10 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## (x + y)2 = x2 + 2xy + y2 for all x, y ∈ R.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 11    PAGE 3

Exercise 1.1 | Q 11 | Page 3

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## Every real number is a complex number.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 12   PAGE 3

Exercise 1.1 | Q 12 | Page 3

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## 1 is a prime number.

### SOLUTION

It is a statement which is false. Hence, it’s truth value is F.

#### EXERCISE 1.1Q 13   PAGE 3

Exercise 1.1 | Q 13 | Page 3

# State which of the following sentences is a statement. Justify your answer if it is a statement. Write down its truth value.

## With the sunset the day ends.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 14    PAGE 3

Exercise 1.1 | Q 14 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## 1 ! = 0

### SOLUTION

It is a statement which is false. Hence, it’s truth value is F.

[Note: Answer in the textbook is incorrect]

#### EXERCISE 1.1Q 15    PAGE 3

Exercise 1.1 | Q 15 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## 3 + 5 > 11

### SOLUTION

It is a statement which is false. Hence, it’s truth value is F.

#### EXERCISE 1.1Q 16   PAGE 3

Exercise 1.1 | Q 16 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## The number π is an irrational number.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

[Note: Answer in the textbook is incorrect]

#### EXERCISE 1.1Q 17   PAGE 3

Exercise 1.1 | Q 17 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## x2 - y2 = (x + y)(x - y) for all x, y ∈ R.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 18    PAGE 3

Exercise 1.1 | Q 18 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## The number 2 is the only even prime number.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 19   PAGE 3

Exercise 1.1 | Q 19 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## Two co-planar lines are either parallel or intersecting.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 20   PAGE 3

Exercise 1.1 | Q 20 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## The number of arrangements of 7 girls in a row for a photograph is 7!.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 21   PAGE 3

Exercise 1.1 | Q 21 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## Give me a compass box.

### SOLUTION

It is an imperative sentence. Hence, it is not a statement.

#### EXERCISE 1.1Q 22   PAGE 3

Exercise 1.1 | Q 22 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## Bring the motor car here.

### SOLUTION

It is an imperative sentence. Hence, it is not a statement.

#### EXERCISE 1.1Q 23    PAGE 3

Exercise 1.1 | Q 23 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## It may rain today.

### SOLUTION

It is an open sentence. Hence, it is not a statement.

#### EXERCISE 1.1Q 24   PAGE 3

Exercise 1.1 | Q 24 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## If a + b < 7, where a ≥ 0 and b ≥ 0 then a < 7 and b < 7.

### SOLUTION

It is a statement which is true. Hence, it’s truth value is T.

#### EXERCISE 1.1Q 25   PAGE 3

Exercise 1.1 | Q 25 | Page 3

# State which of the following sentence is a statement. Justify your answer if it is a statement. Write down its truth value.

## Can you speak in English?

### It is an interrogative sentence. Hence, it is not a statement.

EXERCISE 1.2 [PAGE 6]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.2 [Page 6]

#### EXERCISE 1.2Q 1.1   PAGE 6

Exercise 1.2 | Q 1.1 | Page 6

Express the following statement in symbolic form.

e is a vowel or 2 + 3 = 5

#### SOLUTION

Let p : e is a vowel.

q : 2 + 3 = 5

The symbolic form is p ∨ q.

#### EXERCISE 1.2Q 1.2   PAGE 6

Exercise 1.2 | Q 1.2 | Page 6

Express the following statement in symbolic form.

Mango is a fruit but potato is a vegetable.

#### SOLUTION

Let p : Mango is a fruit.

q : Potato is a vegetable.

The symbolic form is p Λ q.

#### EXERCISE 1.2Q 1.3   PAGE 6

Exercise 1.2 | Q 1.3 | Page 6

Express the following statement in symbolic form.

Milk is white or grass is green.

#### SOLUTION

Let p : Milk is white.

q : Grass is green.

The symbolic form is p ∨ q.

#### EXERCISE 1.2Q 1.4   PAGE 6

Exercise 1.2 | Q 1.4 | Page 6

Express the following statement in symbolic form.

I like playing but not singing.

#### SOLUTION

Let p : I like playing.

q : I do not like singing.

The symbolic form is p ∧ q.

#### EXERCISE 1.2Q 1.5   PAGE 6

Exercise 1.2 | Q 1.5 | Page 6

Express the following statement in symbolic form.

Even though it is cloudy, it is still raining.

#### SOLUTION

Let p : It is cloudy.

q : It is still raining.

The symbolic form is p ∧ q.

#### EXERCISE 1.2Q 2.1   PAGE 6

Exercise 1.2 | Q 2.1 | Page 6

Write the truth value of the following statement.

Earth is a planet and Moon is a star.

#### SOLUTION

Let p : Earth is a planet.

q : Moon is a star.

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

The given statement in symbolic form is p ∧ q.

∴ p ∧ q ≡ T ∧ F ≡ F

∴ Truth value of the given statement is F.

#### EXERCISE 1.2Q 2.2   PAGE 6

Exercise 1.2 | Q 2.2 | Page 6

Write the truth value of the following statement.

16 is an even number and 8 is a perfect square.

#### SOLUTION

Let p : 16 is an even number.

q : 8 is a perfect square.

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

The given statement in symbolic form is p ∧ q.

∴ p ∧ q ≡ T ∧ F ≡ F

∴ Truth value of the given statement is F.

#### EXERCISE 1.2Q 2.3   PAGE 6

Exercise 1.2 | Q 2.3 | Page 6

Write the truth value of the following statement.

A quadratic equation has two distinct roots or 6 has three prime factors.

#### SOLUTION

Let p : A quadratic equation has two distinct roots.

q : 6 has three prime factors.

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

The given statement in symbolic form is p ∨ q.

∴ p ∨ q ≡ F ∨ F ≡ F

∴ Truth value of the given statement is F.

#### EXERCISE 1.2Q 2.4   PAGE 6

Exercise 1.2 | Q 2.4 | Page 6

Write the truth value of the following statement.

The Himalayas are the highest mountains but they are part of India in the North East.

#### SOLUTION

Let p : Himalayas are the highest mountains.

q : Himalayas are the part of India in the north east.

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

The given statement in symbolic form is p ∧ q.

∴ p ∧ q ≡ T ∧ T ≡ T

∴ Truth value of the given statement is T.

EXERCISE 1.3PAGE 7

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.3 [Page 7]

#### EXERCISE 1.3Q 1.1  PAGE 7

Exercise 1.3 | Q 1.1 | Page 7

Write the negation of the following statement.

All men are animals.

#### SOLUTION

Some men are not animals.

#### EXERCISE 1.3Q 1.2  PAGE 7

Exercise 1.3 | Q 1.2 | Page 7

Write the negation of the following statement.

− 3 is a natural number.

#### SOLUTION

– 3 is not a natural number.

#### EXERCISE 1.3Q 1.3  PAGE 7

Exercise 1.3 | Q 1.3 | Page 7

Write the negation of the following statement.

It is false that Nagpur is capital of Maharashtra

#### SOLUTION

Nagpur is capital of Maharashtra.

#### EXERCISE 1.3Q 1.4  PAGE 7

Exercise 1.3 | Q 1.4 | Page 7

Write the negation of the following statement.

2 + 3 ≠ 5

2 + 3 = 5

#### EXERCISE 1.3Q 2.1  PAGE 7

Exercise 1.3 | Q 2.1 | Page 7

Write the truth value of the negation of the following statement.

$5$

## √5 is an irrational number.

#### SOLUTION

Truth value of the given statement is T.

∴ Truth value of its negation is F.

#### EXERCISE 1.3Q 2.2  PAGE 7

Exercise 1.3 | Q 2.2 | Page 7

Write the truth value of the negation of the following statement.

London is in England.

#### SOLUTION

Truth value of the given statement is T.

∴ Truth value of its negation is F.

#### EXERCISE 1.3Q 2.3  PAGE 7

Exercise 1.3 | Q 2.3 | Page 7

Write the truth value of the negation of the following statement.

For every x ∈ N, x + 3 < 8.

#### SOLUTION

Truth value of the given statement is F.

∴ Truth value of its negation is T.

EXERCISE 1.4PAGES 10 - 11

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.4 [Pages 10 - 11]

EXERCISE 1.4Q 1.1    PAGE 10
Exercise 1.4 | Q 1.1 | Page 10

Write the following statement in symbolic form.

If triangle is equilateral then it is equiangular.

#### SOLUTION

Let p : Triangle is equilateral.

q : Triangle is equiangular.

The symbolic form is p → q.

EXERCISE 1.4Q 1.2   PAGE 10
Exercise 1.4 | Q 1.2 | Page 10

Write the following statement in symbolic form.

It is not true that “i” is a real number.

#### SOLUTION

Let p : i is a real number.

The symbolic form is ~ p.

EXERCISE 1.4Q 1.3   PAGE 10
Exercise 1.4 | Q 1.3 | Page 10

Write the following statement in symbolic form.

Even though it is not cloudy, it is still raining.

#### SOLUTION

Let p : It is cloudy.

q : It is raining.

The symbolic form is ~p ∧ q.

EXERCISE 1.4Q 1.4   PAGE 10
Exercise 1.4 | Q 1.4 | Page 10

Write the following statement in symbolic form.

Milk is white if and only if the sky is not blue.

#### SOLUTION

Let p : Milk is white.

q : Sky is blue.

The symbolic form is p ↔ ~ q.

EXERCISE 1.4Q 1.5   PAGE 10
Exercise 1.4 | Q 1.5 | Page 10

Write the following statement in symbolic form.

Stock prices are high if and only if stocks are rising.

#### SOLUTION

Let p : Stock prices are high.

q : Stock are rising

The symbolic form is p ↔ q.

EXERCISE 1.4Q 1.6   PAGE 10
Exercise 1.4 | Q 1.6 | Page 10

Write the following statement in symbolic form.

If Kutub-Minar is in Delhi then Taj-Mahal is in Agra.

#### SOLUTION

Let p : Kutub-Minar is in Delhi.

q : Taj-Mahal Is in Agra.

The symbolic form is p → q.

EXERCISE 1.4Q 2.1    PAGE 11
Exercise 1.4 | Q 2.1 | Page 11

Find the truth value of the following statement.

It is not true that 3 − 7i is a real number.

#### SOLUTION

Let p : 3 – 7i is a real number.

The truth value of p is F.

The given statement in symbolic form is ~p.

∴ ~ p ≡ ~ F ≡ T

∴ Truth value of the given statement is T.

EXERCISE 1.4Q 2.2    PAGE 11
Exercise 1.4 | Q 2.2 | Page 11

Find the truth value of the following statement.

If a joint venture is a temporary partnership, then discount on purchase is credited to the supplier.

#### SOLUTION

Let p : A joint venture is a temporary partnership. q : Discount on purchase is credited to the supplier.

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

The given statement in symbolic form is p → q.

∴ p → q ≡ T → F ≡ F

∴ Truth value of the given statement is F.

EXERCISE 1.4Q 2.3   PAGE 11
Exercise 1.4 | Q 2.3 | Page 11

Find the truth value of the following statement.

Every accountant is free to apply his own accounting rules if and only if machinery is an asset.

#### SOLUTION

Let p : Every accountant is free to apply his own accounting rules.

q : Machinery is an asset.

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

The given statement in symbolic form is p ↔ q.

∴ p ↔ q ≡ F ↔ T ≡ F

∴ Truth value of the given statement is F.

EXERCISE 1.4Q 2.4    PAGE 11
Exercise 1.4 | Q 2.4 | Page 11

Find the truth value of the following statement.

Neither 27 is a prime number nor divisible by 4.

#### SOLUTION

Let p : 27 is a prime number.

q : 27 is divisible by 4.

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

The given statement in symbolic form is ~ p ∧ ~ q.

∴ ~ p ∧ ~ q ≡ ~ F ∧ ~ F ≡ T ∧ T ≡ T

∴ Truth value of the given statement is T.

EXERCISE 1.4Q 2.5    PAGE 11
Exercise 1.4 | Q 2.5 | Page 11

Find the truth value of the following statement.

3 is a prime number and an odd number.

#### SOLUTION

Let p : 3 is a prime number.

q : 3 is an odd number.

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

The given statement in symbolic form is p ∧ q.

∴ p ∧ q ≡ T ∧ T ≡ T

∴ Truth value of the given statement is T.

EXERCISE 1.4Q 3.1    PAGE 11
Exercise 1.4 | Q 3.1 | Page 11

If p and q are true and r and s are false, find the truth value of the following compound statement.

p ∧ (q ∧ r)

#### SOLUTION

p ∧ (q ∧ r) ≡ T ∧ (T ∧ F)

≡ T ∧ F

≡ F

Hence, truth value if F.

EXERCISE 1.4Q 3.2    PAGE 11

Exercise 1.4 | Q 3.2 | Page 11

If p and q are true and r and s are false, find the truth value of the following compound statement.

(p → q) ∨ (r ∧ s)

#### SOLUTION

(p → q) ∨ (r ∧ s) ≡ (T → T) ∨ (F ∧ F)

≡ T ∨ F

≡ T
Hence, truth value if T.

EXERCISE 1.4Q 3.3    PAGE 11
Exercise 1.4 | Q 3.3 | Page 11

If p and q are true and r and s are false, find the truth value of the following compound statement.

~ [(~ p ∨ s) ∧ (~ q ∧ r)]

#### SOLUTION

~ [(~ p ∨ s) ∧ (~ q ∧ r)] ≡ ~[(~T ∨ F) ∧ (~T ∧ F)]

≡ ~[(F ∨ F) ∧ (F ∧ F)

≡ ~ (F ∧ F)

≡ ~ F

≡ T
Hence, truth value if T.

EXERCISE 1.4Q 3.4    PAGE 11
Exercise 1.4 | Q 3.4 | Page 11

If p and q are true and r and s are false, find the truth value of the following compound statement.

(p → q) ↔ ~(p ∨ q)

#### SOLUTION

(p → q) ↔ ~(p ∨ q) ≡ (T → T) ↔ (T ∨ T)

≡ T ↔ ~ T

≡ T ↔ F

≡ F

Hence, truth value if F.

EXERCISE 1.4Q 3.5    PAGE 11
Exercise 1.4 | Q 3.5 | Page 11

If p and q are true and r and s are false, find the truth value of the following compound statement.

[(p ∨ s) → r] ∨ ~ [~ (p → q) ∨ s]

#### SOLUTION

[(p ∨ s) → r] ∨ ~ [~ (p → q) ∨ s]

≡ [(T ∨ F) → F] ∨ ~[~ (T → T) ∨ F]

≡ (T → F) ∨ ~ (~ T ∨ F)

≡ F ∨ ~ (F ∨ F)

≡ F ∨ ~ F

≡ F ∨ T

≡ T

Hence, truth value is T.

EXERCISE 1.4Q 3.6   PAGE 11
Exercise 1.4 | Q 3.6 | Page 11

If p and q are true and r and s are false, find the truth value of the following compound statement.

~ [p ∨ (r ∧ s)] ∧ ~ [(r ∧ ~ s) ∧ q]

#### SOLUTION

~ [p ∨ (r ∧ s)] ∧ ~ [(r ∧ ~ s) ∧ q]

≡ ~ [T ∨ (F ∧ F)] ∧ ~ [(F ∧ ~ F) ∧ T]

≡ ~ (T ∨ F) ∧ ~ [(F ∧ T) ∧ T]

≡ ~ T ∧ ~ (F ∧ T)

≡ F ∧ ~ F

≡ F ∧ T

≡ F

Hence, truth value is F.

EXERCISE 1.4Q 4.1    PAGE 11
Exercise 1.4 | Q 4.1 | Page 11

Assuming that the following statement is true,

p : Sunday is holiday,

q : Ram does not study on holiday,

find the truth values of the following statements.

Sunday is not holiday or Ram studies on holiday.

#### SOLUTION

Symbolic form of the given statement is ~ p ∨~q

∴ ~ p ∨~ q ≡ ~ T ∨ ~ T

≡ F ∨ F

≡ F

Hence, truth value is F.

EXERCISE 1.4Q 4.2    PAGE 11
Exercise 1.4 | Q 4.2 | Page 11

Assuming that the following statement is true,

p : Sunday is holiday,

q : Ram does not study on holiday,

find the truth values of the following statements.

If Sunday is not holiday then Ram studies on holiday.

#### SOLUTION

Symbolic form of the given statement is
~ p → ~ q

∴ ~ p → ~ q ≡ ~ T → ~ T

≡ F → F

≡ T

Hence, truth value is T.

EXERCISE 1.4Q 4.3    PAGE 11
Exercise 1.4 | Q 4.3 | Page 11

Assuming that the following statement is true,

p : Sunday is holiday,

q : Ram does not study on holiday,

find the truth values of the following statements.

Sunday is a holiday and Ram studies on holiday.

#### SOLUTION

Symbolic form of the given statement is p ∧ ~ q

∴ p ∧ ~ q ≡ T ∧ ~ T

≡ T ∧ F

≡ F

Hence, truth value is F.

EXERCISE 1.4Q 5.1    PAGE 11
Exercise 1.4 | Q 5.1 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

p ↔ ~ q

#### SOLUTION

He swims if and only if water is not warm.

EXERCISE 1.4Q 5.2   PAGE 11
Exercise 1.4 | Q 5.2 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

~ (p ∨ q)

#### SOLUTION

It is not true that he swims or water is warm.

EXERCISE 1.4Q 5.3    PAGE 11
Exercise 1.4 | Q 5.3 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

q → p

#### SOLUTION

If water is warm then he swims.

EXERCISE 1.4Q 5.4    PAGE 11
Exercise 1.4 | Q 5.4 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

q ∧ ~ p

#### SOLUTION

Water is warm and he does not swim.

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.5 [Page 12]

EXERCISE 1.5Q 1.1    PAGE 12

Exercise 1.5 | Q 1.1 | Page 12

Use quantifiers to convert the following open sentences defined on N, into a true statement.

x2 + 3x - 10 = 0

#### SOLUTION

∃ x ∈ N, such that x2 + 3x – 10 = 0

It is true statement, since x = 2 ∈ N satisfies it.

EXERCISE 1.5Q 1.2   PAGE 12
Exercise 1.5 | Q 1.2 | Page 12

Use quantifiers to convert the following open sentences defined on N, into a true statement.

3x - 4 < 9

#### SOLUTION

∃ x ∈ N, such that 3x – 4 < 9

It is true statement, since

x = 2, 3, 4 ∈ N satisfies 3x - 4 < 9.

EXERCISE 1.5Q 1.3    PAGE 12
Exercise 1.5 | Q 1.3 | Page 12

Use quantifiers to convert the following open sentences defined on N, into a true statement.

n2 ≥ 1

#### SOLUTION

∀ n ∈ N, n2 ≥ 1

It is true statement, since all n ∈ N satisfy it.

EXERCISE 1.5Q 1.4    PAGE 12
Exercise 1.5 | Q 1.4 | Page 12

Use quantifiers to convert the following open sentences defined on N, into a true statement.

2n - 1 = 5

#### SOLUTION

∃ n ∈ N, such that 2n - 1 = 5

It is a true statement since all n = 3 ∈ N satisfy 2n - 1 = 5.

EXERCISE 1.5Q 1.5    PAGE 12
Exercise 1.5 | Q 1.5 | Page 12

Use quantifiers to convert the following open sentences defined on N, into a true statement.

y + 4 > 6

#### SOLUTION

∃ y ∈ N, such that y + 4 > 6

It is a true statement since y = 3, 4, ... ∈ N satisfy y + 4 > 6.

EXERCISE 1.5Q 1.6    PAGE 12
Exercise 1.5 | Q 1.6 | Page 12

Use quantifiers to convert the following open sentences defined on N, into a true statement.

3y - 2 ≤ 9

#### SOLUTION

∃ y ∈ N, such that 3y - 2 ≤ 9

It is a true statement since y = 1, 2, 3 ∈ N satisfy it.

EXERCISE 1.5Q 2.1    PAGE 12
Exercise 1.5 | Q 2.1 | Page 12

If B = {2, 3, 5, 6, 7} determine the truth value of ∀ x ∈ B such that x is prime number.

#### SOLUTION

For x = 6, x is not a prime number.

∴ x = 6 does not satisfies the given statement.

∴ The given statement is false.

∴ It’s truth value is F.

EXERCISE 1.5Q 2.2    PAGE 12
Exercise 1.5 | Q 2.2 | Page 12

If B = {2, 3, 5, 6, 7} determine the truth value of
∃ n ∈ B, such that n + 6 > 12.

#### SOLUTION

For n = 7, n + 6 = 7 + 6 = 13 > 12

∴ n = 7 satisfies the equation n + 6 > 12.

∴ The given statement is true.

∴ It’s truth value is T.

EXERCISE 1.5Q 2.3    PAGE 12
Exercise 1.5 | Q 2.3 | Page 12

If B = {2, 3, 5, 6, 7} determine the truth value of
∃ n ∈ B, such that 2n + 2 < 4.

#### SOLUTION

There is no n in B which satisfies 2n + 2 < 4.

∴ The given statement is false.

∴ It’s truth value is F.

EXERCISE 1.5Q 2.4    PAGE 12
Exercise 1.5 | Q 2.4 | Page 12

If B = {2, 3, 5, 6, 7} determine the truth value of
∀ y ∈ B, such that y2 is negative.

#### SOLUTION

There is no y in B which satisfies y2 < 0.

∴ The given statement is false.

∴ It’s truth value is F.

EXERCISE 1.5Q 2.5    PAGE 12
Exercise 1.5 | Q 2.5 | Page 12

If B = {2, 3, 5, 6, 7} determine the truth value of
∀ y ∈ B, such that (y - 5) ∈ N

#### SOLUTION

For y = 2, y – 5 = 2 – 5 = –3 ∉ N.

∴ y = 2 does not satisfies the equation (y – 5) ∈ N.

∴ The given statement is false.

∴ It’s truth value is F.

EXERCISE 1.6 [PAGE 16]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.6 [Page 16]

EXERCISE 1.6Q 1.1   PAGE 16
Exercise 1.6 | Q 1.1 | Page 16

Prepare truth tables for the following statement pattern.

p → (~ p ∨ q)

#### SOLUTION

p → (~ p ∨ q)

 p q ~p ~ p ∨ q p → (~ p ∨ q) T T F T T T F F F F F T T T T F F T T T
EXERCISE 1.6Q 1.2   PAGE 16
Exercise 1.6 | Q 1.2 | Page 16

Prepare truth tables for the following statement pattern.

(~ p ∨ q) ∧ (~ p ∨ ~ q)

#### SOLUTION

(~ p ∨ q) ∧ (~ p ∨ ~ q)

 p q ~p ~q ~p∨q ~p∨~q (~p∨q)∧(~p∨~q) T T F F T F F T F F T F T F F T T F T T T F F T T T T T
EXERCISE 1.6Q 1.3   PAGE 16
Exercise 1.6 | Q 1.3 | Page 16

Prepare truth tables for the following statement pattern.

(p ∧ r) → (p ∨ ~ q)

#### SOLUTION

(p ∧ r) → (p ∨ ~ q)

 p q r ~q p ∧ r p∨~q (p ∧ r) → (p ∨ ~ q) T T T F T T T T T F F F T T T F T T T T T T F F T F T T F T T F F F T F T F F F F T F F T T F T T F F F T F T T
EXERCISE 1.6Q 1.4   PAGE 16
Exercise 1.6 | Q 1.4 | Page 16

Prepare truth tables for the following statement pattern.

(p ∧ q) ∨ ~ r

#### SOLUTION

(p ∧ q) ∨ ~ r

 p q r ~r p ∧ q (p ∧ q) ∨ ~ r T T T F T T T T F T T T T F T F F F T F F T F T F T T F F F F T F T F T F F T F F F F F F T F T
EXERCISE 1.6Q 2.1   PAGE 16
Exercise 1.6 | Q 2.1 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

q ∨ [~ (p ∧ q)]

#### SOLUTION

 p q p ∧ q ~ (p ∧ q) q ∨ [~ (p ∧ q)] T T T F T T F F T T F T F T T F F F T T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 2.2   PAGE 16
Exercise 1.6 | Q 2.2 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(~ q ∧ p) ∧ (p ∧ ~ p)

#### SOLUTION

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

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 2.3   PAGE 16
Exercise 1.6 | Q 2.3 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(p ∧ ~ q) → (~ p ∧ ~ q)

#### SOLUTION

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

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 2.4   PAGE 16
Exercise 1.6 | Q 2.4 | Page 16

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

~ p → (p → ~ q)

#### SOLUTION

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

All the truth values in the last column are T. Hence, it is tautology.

EXERCISE 1.6Q 3.1   PAGE 16
Exercise 1.6 | Q 3.1 | Page 16

Prove that the following statement pattern is a tautology.

(p ∧ q) → q

#### SOLUTION

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

All the truth values in the last column are T. Hence, it is tautology.

EXERCISE 1.6Q 3.2   PAGE 16
Exercise 1.6 | Q 3.2 | Page 16

Prove that the following statement pattern is a tautology.

(p → q) ↔ (~ q → ~ p)

#### SOLUTION

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

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 3.3   PAGE 16
Exercise 1.6 | Q 3.3 | Page 16

Prove that the following statement pattern is a tautology.

(~p ∧ ~q ) → (p → q)

#### SOLUTION

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

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 3.4   PAGE 16
Exercise 1.6 | Q 3.4 | Page 16

Prove that the following statement pattern is a tautology.

(~ p ∨ ~ q) ↔ ~ (p ∧ q)

#### SOLUTION

 p q ~p ~q ~p∨~q p∧q ~p∨~q (~p∨~q↔~(p ∧ q) T T F F F T F T T F F T T F T T F T T F T F T T F F T T T F T T

All the truth values in the last column are T. Hence, it is a tautology.

EXERCISE 1.6Q 4.1   PAGE 16
Exercise 1.6 | Q 4.1 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∨ q) ∧ (~p ∧ ~q)

#### SOLUTION

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

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 4.2   PAGE 16
Exercise 1.6 | Q 4.2 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ ~p

#### SOLUTION

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

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 4.3   PAGE 16
Exercise 1.6 | Q 4.3 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ (~p ∨ ~q)

#### SOLUTION

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

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 4.4   PAGE 16
Exercise 1.6 | Q 4.4 | Page 16

Prove that the following statement pattern is a contradiction.

(p → q) ∧ (p ∧ ~ q)

#### SOLUTION

 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 T F F F F T T F F

All the truth values in the last column are F. Hence, it is a contradiction.

EXERCISE 1.6Q 5.1   PAGE 16
Exercise 1.6 | Q 5.1 | Page 16

Show that the following statement pattern is contingency.

(p∧~q) → (~p∧~q)

#### SOLUTION

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

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 5.2   PAGE 16
Exercise 1.6 | Q 5.2 | Page 16

Show that the following statement pattern is contingency.

(p → q) ↔ (~ p ∨ q)

#### SOLUTION

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

All the truth values in the last column are T. Hence, it is a tautology. Not contingency.

EXERCISE 1.6Q 5.3   PAGE 16
Exercise 1.6 | Q 5.3 | Page 16

Show that the following statement pattern is contingency.

p ∧ [(p → ~ q) → q]

#### SOLUTION

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

Truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 5.4   PAGE 16
Exercise 1.6 | Q 5.4 | Page 16

Show that the following statement pattern is contingency.

(p → q) ∧ (p → r)

#### SOLUTION

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

The truth values in the last column are not identical. Hence, it is contingency.

EXERCISE 1.6Q 6.1   PAGE 16

Exercise 1.6 | Q 6.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

#### SOLUTION

 1 2 3 4 5 6 7 8 p q r q∧r p∨(q∧r) p∨q p∨r (p∨q)∧(p∨r) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F

The entries in columns 5 and 8 are identical.

∴ p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

EXERCISE 1.6Q 6.2   PAGE 16
Exercise 1.6 | Q 6.2 | Page 16

Using the truth table, verify

p → (p → q) ≡ ~ q → (p → q)

#### SOLUTION

 1 2 3 4 5 6 p q ~q p→q p→(p→q) ~q→(p→q) T T F T T T T F T F F F F T F T T T F F T T T T

In the above truth table, entries in columns 5 and 6 are identical.

∴ p → (p → q) ≡ ~ q → (p → q)

EXERCISE 1.6Q 6.3   PAGE 16
Exercise 1.6 | Q 6.3 | Page 16

Using the truth table, verify

~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q

#### SOLUTION

 1 2 3 4 5 6 7 8 p q ~q p→~q ~(p→~q) ~(~q) p∧~(~q) p∧q T T F F T T T T T F T T F F F F F T F T F T F F F F T T F F F F

In the above table, entries in columns 5, 7, and 8 are identical.

∴ ~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q

EXERCISE 1.6Q 6.4   PAGE 16
Exercise 1.6 | Q 6.4 | Page 16

Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

#### SOLUTION

 1 2 3 4 5 6 7 p q ~p (p∨q) ~(p∨q) ~p∧q ~(p∨q)∨(~p∧q) T T F T F F F T F F T F F F F T T T F T T F F T F T F T

In the above truth table, the entries in columns 3 and 7 are identical.

∴ ~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

EXERCISE 1.6Q 7.1   PAGE 16

Exercise 1.6 | Q 7.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

#### SOLUTION

 1 2 3 4 5 6 7 8 p q r q∧r p∨(q∧r) p∨q p∨r (p∨q)∧(p∨r) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F

The entries in columns 5 and 8 are identical.

∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

EXERCISE 1.6Q 7.2   PAGE 16
Exercise 1.6 | Q 7.2 | Page 16

Prove that the following pair of statement pattern is equivalent.

p ↔ q and (p → q) ∧ (q → p)

#### SOLUTION

 1 2 3 4 5 6 p q p↔q p→q q→p (p→q)∧(q→p) T T T T T T T F F F T F F T F T F F F F T T T T

In the above table, entries in columns 3 and 6 are identical.

∴ Statement p ↔ q and (p → q) ∧ (q → p) are equivalent.

EXERCISE 1.6Q 7.3   PAGE 16
Exercise 1.6 | Q 7.3 | Page 16

Prove that the following pair of statement pattern is equivalent.

p → q and ~ q → ~ p and ~ p ∨ q

#### SOLUTION

 1 2 3 4 5 6 7 p q ~p ~q p→q ~q→~p ~p∨q T T F F T T T T F F T F F F F T T F T T T F F T T T T T

In the above table, entries in columns 5, 6 and 7 are identical.

∴ Statement p → q and ~q → ~p and ~p ∨ q are equivalent.

EXERCISE 1.6Q 7.4   PAGE 16
Exercise 1.6 | Q 7.4 | Page 16

Prove that the following pair of statement pattern is equivalent.

~(p ∧ q) and ~p ∨ ~q

#### SOLUTION

 1 2 3 4 5 6 7 p q ~p ~q p∧q ~(p∧q) ~p∨~q T T F F T F F T F F T F T T F T T F F T T F F T T F T T

In the above table, entries in columns 6 and 7 are identical.

∴ Statement ~(p ∧ q) and ~p ∨ ~q are equivalent.

EXERCISE 1.7 [PAGE 17]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.7 [Page 17]

EXERCISE 1.7Q 1.1   PAGE 17
Exercise 1.7 | Q 1.1 | Page 17

Write the dual of the following:

(p ∨ q) ∨ r

#### SOLUTION

(p ∧ q) ∧ r

EXERCISE 1.7Q 1.2   PAGE 17
Exercise 1.7 | Q 1.2 | Page 17

Write the dual of the following:

~(p ∨ q) ∧ [p ∨ ~ (q ∧ ~ r)]

#### SOLUTION

~(p ∧ q) ∨ [p ∧ ~ (q ∨ ~ r)]

EXERCISE 1.7Q 1.3   PAGE 17
Exercise 1.7 | Q 1.3 | Page 17

Write the dual of the following:

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r

#### SOLUTION

p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r

EXERCISE 1.7Q 1.4   PAGE 17
Exercise 1.7 | Q 1.4 | Page 17

Write the dual of the following:

~(p ∧ q) ≡ ~ p ∨ ~ q

#### SOLUTION

~(p ∨ q) ≡ ~ p ∧ ~ q

EXERCISE 1.7Q 2.1   PAGE 17
Exercise 1.7 | Q 2.1 | Page 17

Write the dual statement of the following compound statement.

13 is a prime number and India is a democratic country.

#### SOLUTION

13 is a prime number or India is a democratic country.

EXERCISE 1.7Q 2.2   PAGE 17
Exercise 1.7 | Q 2.2 | Page 17

Write the dual statement of the following compound statement.

Karina is very good or everybody likes her.

#### SOLUTION

Karina is very good and everybody likes her.

EXERCISE 1.7Q 2.3   PAGE 17
Exercise 1.7 | Q 2.3 | Page 17

Write the dual statement of the following compound statement.