Chapter 1: Mathematical Logic HSC

Chapter 1: Mathematical Logic

Exercise 1.1

Chapter 1: Mathematical Logic  Exercise 1.1


Chapter 1: Mathematical Logic  Exercise 1.1


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?

SOLUTION

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]

Write the truth value of the following statement.

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.

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.


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

SOLUTION

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.

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
Write the following statement in symbolic form.
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
Find the truth value of the following statement.
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
If p and q are true and r and s are false, find the truth value of the following compound statement.
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
Assuming that the following statement is true,  p : Sunday is holiday,  q : Ram does not study on holiday,
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
If p : He swims  q : Water is warm  Give the verbal statement for the following symbolic statement.
Exercise 1.4 | Q 5.1 | Page 11

If p : He swims

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

If p : He swims

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

If p : He swims

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

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

q ∧ ~ p

SOLUTION

Water is warm and he does not swim.


EXERCISE 1.5PAGE 12

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


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

EXERCISE 1.6PAGE 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)  p → (~ p ∨ q)

Prepare truth tables for the following statement pattern.

p → (~ p ∨ q)

SOLUTION

p → (~ p ∨ q)

pq~p~ p ∨ qp → (~ p ∨ q)
TTFTT
TFFFF
FTTTT
FFTTT
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)

pq~p~q~p∨q~p∨~q(~p∨q)∧(~p∨~q)
TTFFTFF
TFFTFTF
FTTFTTT
FFTTTTT
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)

pqr~qp ∧ rp∨~q(p ∧ r) → (p ∨ ~ q)
TTTFTTT
TTFFFTT
TFTTTTT
TFFTFTT
FTTFFFT
FTFFFFT
FFTTFTT
FFFTFTT
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

pqr~rp ∧ q(p ∧ q) ∨ ~ r
TTTFTT
TTFTTT
TFTFFF
TFFTFT
FTTFFF
FTFTFT
FFTFFF
FFFTFT
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

pqp ∧ q~ (p ∧ q)q ∨ [~ (p ∧ q)]
TTTFT
TFFTT
FTFTT
FFFTT

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)
Examine whether the following statement pattern is a tautology, a contradiction or a contingency.  (~ q ∧ p) ∧ (p ∧ ~ p)

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

(~ q ∧ p) ∧ (p ∧ ~ p)

SOLUTION

pq~p~q(~q∧p)(p∧~p)(~q∧p)∧(p∧~p)
TTFFFFF
TFFTTFF
FTTFFFF
FFTTFFF

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

pq~p~qp∧~q~p∧~q(p∧~q)→(~p∧~q)
TTFFFFT
TFFTTFF
FTTFFFT
FFTTFTT

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

pq~p~qp→~q~p→(p→~q)
TTFFFT
TFFTTT
FTTFTT
FFTTTT

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

Prove that the following statement pattern is a tautology.

(p ∧ q) → q

SOLUTION

pqp ∧ q(p∧q)→q
TTTT
TFFT
FTFT
FFFT

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

pq~p~qp→q~q→~p(p→q)↔(~q→~p)
TTFFTTT
TFFTFFT
FTTFTTT
FFTTTTT

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

pq~p~q~p∧~qp→q(~p∧~q)→(p→q)
TTFFFTT
TFFTFFT
FTTFFTT
FFTTTTT

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

pq~p~q~p∨~qp∧q~p∨~q(~p∨~q↔~(p ∧ q)
TTFFFTFT
TFFTTFTT
FTTFTFTT
FFTTTFTT

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)

Prove that the following statement pattern is a contradiction.

(p ∨ q) ∧ (~p ∧ ~q)

SOLUTION

pq~p~qp∨q~p∧~q(p∨q)∧(~p∧~q)
TTFFTFF
TFFTTFF
FTTFTFF
FFTTFTF

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

pq~pp∧q(p∧q)∧~p
TTFTF
TFFFF
FTTFF
FFTFF

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

pq~p~qp∧q~p∨~q(p∧q)∧(~p∨~q)
TTFFTFF
TFFTFTF
FTTFFTF
FFTTFTF

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

pq~qp→qp∧~q(p→q)∧(p∧~q)
TTFTFF
TFTFTF
FTFTFF
FFTTFF

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)

Show that the following statement pattern is contingency.

(p∧~q) → (~p∧~q)

SOLUTION

pq~p~qp∧~q~p∧~q(p∧~q)→(~p∧~q)
TTFFFFT
TFFTTFF
FTTFFFT
FFTTFTT

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

pq~pp→q~p∨q(p→q)↔(~p∨q)
TTFTTT
TFFFFT
FTTTTT
FFTTTT

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

pq~qp→~q(p→~q)→qp∧[(p→~q)→q]
TTFFTT
TFTTFF
FTFTTF
FFTTFF

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

pqrp→qp→r(p→q)∧(p→r)
TTTTTT
TTFTFF
TFTFTF
TFFFFF
FTTTTT
FTFTTT
FFTTTT
FFFTTT

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

EXERCISE 1.6Q 6.1   PAGE 16
Using the truth table, verify  p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Exercise 1.6 | Q 6.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

SOLUTION

12345678
pqrq∧rp∨(q∧r)p∨qp∨r(p∨q)∧(p∨r)
TTTTTTTT
TTFFTTTT
TFTFTTTT
TFFFTTTT
FTTTTTTT
FTFFFTFF
FFTFFFTF
FFFFFFFF

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

123456
pq~qp→qp→(p→q)~q→(p→q)
TTFTTT
TFTFFF
FTFTTT
FFTTTT

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

12345678
pq~qp→~q

~(p→~q)

~(~q)p∧~(~q)p∧q
TTFFTTTT
TFTTFFFF
FTFTFTFF
FFTTFFFF

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

1234567
pq~p(p∨q)~(p∨q)~p∧q~(p∨q)∨(~p∧q)
TTFTFFF
TFFTFFF
FTTTFTT
FFTFTFT

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
Using the truth table, verify  p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Exercise 1.6 | Q 7.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

SOLUTION

12345678
pqrq∧rp∨(q∧r)p∨qp∨r(p∨q)∧(p∨r)
TTTTTTTT
TTFFTTTT
TFTFTTTT
TFFFTTTT
FTTTTTTT
FTFFFTFF
FFTFFFTF
FFFFFFFF

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

123456
pqp↔qp→qq→p(p→q)∧(q→p)
TTTTTT
TFFFTF
FTFTFF
FFTTTT

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

1234567
pq~p~qp→q~q→~p~p∨q
TTFFTTT
TFFTFFF
FTTFTTT
FFTTTTT

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

1234567
pq~p~qp∧q~(p∧q)~p∨~q
TTFFTFF
TFFTFTT
FTTFFTT
FFTTFTT

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]

EXERCISE 1.7PAGE 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.

Radha and Sushmita cannot read Urdu.

SOLUTION

Radha or Sushmita cannot read Urdu.

EXERCISE 1.7Q 2.4   PAGE 17
Exercise 1.7 | Q 2.4 | Page 17

Write the dual statement of the following compound statement.

A number is a real number and the square of the number is non-negative.

SOLUTION

A number is a real number or the square of the number is non-negative.

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