# Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 1 - Mathematical Logic [Latest edition]

## Chapter 1: Mathematical Logic

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Exercise 1.6Exercise 1.7Exercise 1.8Exercise 1.9Exercise 1.10Miscellaneous Exercise 1
Exercise 1.1 [Pages 2 - 3]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.1 [Pages 2 - 3]

Exercise 1.1 | Q 1 | Page 2

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

A triangle has ‘n’ sides

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°

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!

Exercise 1.1 | Q 4 | Page 2

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

Exercise 1.1 | Q 5 | Page 2

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

sqrt(-4) is an irrational number.

Exercise 1.1 | Q 6 | Page 2

State which of the following sentence 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.

Exercise 1.1 | Q 7 | Page 3

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

He is an actor.

Exercise 1.1 | Q 8 | Page 3

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

Did you eat lunch yet?

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.

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.

Exercise 1.1 | Q 11 | Page 3

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

Every real number is a complex number.

Exercise 1.1 | Q 12 | 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 is a prime number.

Exercise 1.1 | Q 13 | Page 3

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

With the sunset the day ends.

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

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

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.

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.

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.

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.

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!.

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.

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.

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.

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.

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?

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 1.2 | Q 1.1 | Page 6

Express the following statement in symbolic form.

e is a vowel or 2 + 3 = 5

Exercise 1.2 | Q 1.2 | Page 6

Express the following statement in symbolic form.

Mango is a fruit but potato is a vegetable.

Exercise 1.2 | Q 1.3 | Page 6

Express the following statement in symbolic form.

Milk is white or grass is green.

Exercise 1.2 | Q 1.4 | Page 6

Express the following statement in symbolic form.

I like playing but not singing.

Exercise 1.2 | Q 1.5 | Page 6

Express the following statement in symbolic form.

Even though it is cloudy, it is still raining.

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.

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.

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.

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.

Exercise 1.3 [Page 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 1.3 | Q 1.1 | Page 7

Write the negation of the following statement.

All men are animals.

Exercise 1.3 | Q 1.2 | Page 7

Write the negation of the following statement.

− 3 is a natural number.

Exercise 1.3 | Q 1.3 | Page 7

Write the negation of the following statement.

It is false that Nagpur is capital of Maharashtra

Exercise 1.3 | Q 1.4 | Page 7

Write the negation of the following statement.

2 + 3 ≠ 5

Exercise 1.3 | Q 2.1 | Page 7

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

sqrt5 is an irrational number.

Exercise 1.3 | Q 2.2 | Page 7

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

London is in England.

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.

Exercise 1.4 [Pages 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 1.4 | Q 1.1 | Page 10

Write the following statement in symbolic form.

If triangle is equilateral then it is equiangular.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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)

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)

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)]

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)

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]

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]

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.

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.

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.

Exercise 1.4 | Q 5.1 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

p ↔ ~ q

Exercise 1.4 | Q 5.2 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

~ (p ∨ q)

Exercise 1.4 | Q 5.3 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

q → p

Exercise 1.4 | Q 5.4 | Page 11

q : Water is warm

Give the verbal statement for the following symbolic statement.

q ∧ ~ p

Exercise 1.5 [Page 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 1.5 | Q 1.1 | Page 12

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

x2 + 3x - 10 = 0

Exercise 1.5 | Q 1.2 | Page 12

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

3x - 4 < 9

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

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

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

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

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.

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.

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.

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.

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

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 1.6 | Q 1.1 | Page 16

Prepare truth tables for the following statement pattern.

p → (~ p ∨ q)

Exercise 1.6 | Q 1.2 | Page 16

Prepare truth tables for the following statement pattern.

(~ p ∨ q) ∧ (~ p ∨ ~ q)

Exercise 1.6 | Q 1.3 | Page 16

Prepare truth tables for the following statement pattern.

(p ∧ r) → (p ∨ ~ q)

Exercise 1.6 | Q 1.4 | Page 16

Prepare truth table for (p ˄ q) ˅ ~ r

(p ∧ q) ∨ ~ r

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)]

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)

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)

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)

Exercise 1.6 | Q 3.1 | Page 16

Prove that the following statement pattern is a tautology.

(p ∧ q) → q

Exercise 1.6 | Q 3.2 | Page 16

Prove that the following statement pattern is a tautology.

(p → q) ↔ (~ q → ~ p)

Exercise 1.6 | Q 3.3 | Page 16

Prove that the following statement pattern is a tautology.

(~p ∧ ~q ) → (p → q)

Exercise 1.6 | Q 3.4 | Page 16

Prove that the following statement pattern is a tautology.

(~ p ∨ ~ q) ↔ ~ (p ∧ q)

Exercise 1.6 | Q 4.1 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∨ q) ∧ (~p ∧ ~q)

Exercise 1.6 | Q 4.2 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ ~p

Exercise 1.6 | Q 4.3 | Page 16

Prove that the following statement pattern is a contradiction.

(p ∧ q) ∧ (~p ∨ ~q)

Exercise 1.6 | Q 4.4 | Page 16

Prove that the following statement pattern is a contradiction.

(p → q) ∧ (p ∧ ~ q)

Exercise 1.6 | Q 5.1 | Page 16

Show that the following statement pattern is contingency.

(p∧~q) → (~p∧~q)

Exercise 1.6 | Q 5.2 | Page 16

Show that the following statement pattern is contingency.

(p → q) ↔ (~ p ∨ q)

Exercise 1.6 | Q 5.3 | Page 16

Show that the following statement pattern is contingency.

p ∧ [(p → ~ q) → q]

Exercise 1.6 | Q 5.4 | Page 16

Show that the following statement pattern is contingency.

(p → q) ∧ (p → r)

Exercise 1.6 | Q 6.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

Exercise 1.6 | Q 6.2 | Page 16

Using the truth table, verify

p → (p → q) ≡ ~ q → (p → q)

Exercise 1.6 | Q 6.3 | Page 16

Using the truth table, verify

~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q

Exercise 1.6 | Q 6.4 | Page 16

Using the truth table, verify

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

Exercise 1.6 | Q 7.1 | Page 16

Using the truth table, verify

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

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)

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

Exercise 1.6 | Q 7.4 | Page 16

Prove that the following pair of statement pattern is equivalent.

~(p ∧ q) and ~p ∨ ~q

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 1.7 | Q 1.1 | Page 17

Write the dual of the following:

(p ∨ q) ∨ r

Exercise 1.7 | Q 1.2 | Page 17

Write the dual of the following:

~(p ∨ q) ∧ [p ∨ ~ (q ∧ ~ r)]

Exercise 1.7 | Q 1.3 | Page 17

Write the dual of the following:

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r

Exercise 1.7 | Q 1.4 | Page 17

Write the dual of the following:

~(p ∧ q) ≡ ~ p ∨ ~ q

Exercise 1.7 | Q 2.1 | Page 17

Write the dual statement of the following compound statement.

13 is prime number and India is a democratic country.

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.

Exercise 1.7 | Q 2.3 | Page 17

Write the dual statement of the following compound statement.

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.

Exercise 1.8 [Page 21]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.8 [Page 21]

Exercise 1.8 | Q 1.1 | Page 21

Write the negation of the following statement.

All the stars are shining if it is night.

Exercise 1.8 | Q 1.2 | Page 21

Write the negation of the following statement.

∀ n ∈ N, n + 1 > 0

Exercise 1.8 | Q 1.3 | Page 21

Write the negation of the following statement.

∃ n ∈ N, (n2 + 2) is odd number.

Exercise 1.8 | Q 1.4 | Page 21

Write the negation of the following statement.

Some continuous functions are differentiable.

Exercise 1.8 | Q 2.1 | Page 21

Using the rules of negation, write the negation of the following:

(p → r) ∧ q

Exercise 1.8 | Q 2.2 | Page 21

Using the rules of negation, write the negation of the following:

~(p ∨ q) → r

Exercise 1.8 | Q 2.3 | Page 21

Using the rules of negation, write the negation of the following:

(~p ∧ q) ∧ (~q ∨ ~r)

Exercise 1.8 | Q 3.1 | Page 21

Write the converse, inverse, and contrapositive of the following statement.

"If it snows, then they do not drive the car"

Exercise 1.8 | Q 3.2 | Page 21

Write the converse, inverse, and contrapositive of the following statement.

If he studies, then he will go to college.

Exercise 1.8 | Q 4.1 | Page 21

With proper justification, state the negation of the following.

(p → q) ∨ (p → r)

Exercise 1.8 | Q 4.2 | Page 21

With proper justification, state the negation of the following.

(p ↔ q) v (~ q → ~r)

Exercise 1.8 | Q 4.3 | Page 21

With proper justification, state the negation of the following.

(p → q) ∧ r

Exercise 1.9 [Page 22]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.9 [Page 22]

Exercise 1.9 | Q 1.1 | Page 22

Without using truth table, show that

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

Exercise 1.9 | Q 1.2 | Page 22

Without using truth table, show that

p ∧ [(~ p ∨ q) ∨ ~ q] ≡ p

Exercise 1.9 | Q 1.3 | Page 22

Without using truth table, show that

~ [(p ∧ q) → ~ q] ≡ p ∧ q

Exercise 1.9 | Q 1.4 | Page 22

Without using truth table, show that

~r → ~ (p ∧ q) ≡ [~ (q → r)] → ~ p

Exercise 1.9 | Q 1.5 | Page 22

Without using truth table, show that

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

Exercise 1.9 | Q 2.1 | Page 22

Using the algebra of statement, prove that

[p ∧ (q ∨ r)] ∨ [~ r ∧ ~ q ∧ p] ≡ p

Exercise 1.9 | Q 2.2 | Page 22

Using the algebra of statement, prove that

(p ∧ q) ∨ (p ∧ ~ q) ∨ (~ p ∧ ~ q) ≡ (p ∨ ~ q)

Exercise 1.9 | Q 2.3 | Page 22

Using the algebra of statement, prove that

(p ∨ q) ∧ (~ p ∨ ~ q) ≡ (p ∧ ~ q) ∨ (~ p ∧ q)

Exercise 1.10 [Page 27]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Exercise 1.10 [Page 27]

Exercise 1.10 | Q 1.1 | Page 27

Represent the truth of the following statement by the Venn diagram.

Some hardworking students are obedient.

Exercise 1.10 | Q 1.2 | Page 27

Represent the truth of the following statement by the Venn diagram.

No circles are polygons.

Exercise 1.10 | Q 1.3 | Page 27

Represent the truth of the following statement by the Venn diagram.

All teachers are scholars and scholars are teachers.

Exercise 1.10 | Q 1.4 | Page 22

Represent the truth of the following statement by the Venn diagram.

If a quadrilateral is a rhombus, then it is a parallelogram.

Exercise 1.10 | Q 2.1 | Page 27

Draw a Venn diagram for the truth of the following statement.

Some share brokers are chartered accountants.

Exercise 1.10 | Q 2.2 | Page 27

Draw a Venn diagram for the truth of the following statement.

No wicket keeper is bowler, in a cricket team.

Exercise 1.10 | Q 3.1 | Page 27

Represent the following statement by the Venn diagram.

Some non-resident Indians are not rich.

Exercise 1.10 | Q 3.2 | Page 27

Represent the following statement by the Venn diagram.

No circle is rectangle.

Exercise 1.10 | Q 3.3 | Page 27

Represent the following statement by the Venn diagram.

If n is a prime number and n ≠ 2, then it is odd.

Miscellaneous Exercise 1 [Pages 29 - 34]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical Logic Miscellaneous Exercise 1 [Pages 29 - 34]

Miscellaneous Exercise 1 | Q 1.01 | Page 29

Choose the correct alternative :

Which of the following is not a statement?

• Smoking is injuries to health

• 2 + 2 = 4

• 2 is the only even prime number.

• Come here

Miscellaneous Exercise 1 | Q 1.02 | Page 29

Choose the correct alternative :

Which of the following is an open statement?

• x is a natural number.

• Give answer a glass of water.

• WIsh you best of luck.

• Good morning to all.

Miscellaneous Exercise 1 | Q 1.03 | Page 29

Choose the correct alternative :

Let p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r). Then, this law is known as.

• commutative law

• associative law

• De-Morgan's law

• distributive law

Miscellaneous Exercise 1 | Q 1.04 | Page 29

Choose the correct alternative :

The false statement in the following is

• p ∧ (∼ p) is contradiction

• (p → q) ↔ (∼ q → ∼ p) is a contradiction.

• ~ (∼ p) ↔ p is a tautology

• p ∨ (∼ p) ↔ p is a tautology

Miscellaneous Exercise 1 | Q 1.05 | Page 29

Choose the correct alternative :

For the following three statements
p : 2 is an even number.
q : 2 is a prime number.
r : Sum of two prime numbers is always even.
Then, the symbolic statement (p ∧ q) → ∼ r means.

• 2 is an even and prime number and the sum of two prime numbers is always even.

• 2 is an even and prime number and the sum of two prime numbers is not always even.

• If 2 is an even and prime number, then the sum of two prime numbers is not always even.

• If 2 is an even and prime number, then the sum of two prime numbers is also even.

Miscellaneous Exercise 1 | Q 1.06 | Page 30

Choose the correct alternative :

If p : He is intelligent.
q : He is strong
Then, symbolic form of statement “It is wrong that, he is intelligent or strong” is

• ∼ p ∨ ∼ p

• ∼ (p ∧ q)

• ∼ (p ∨ q)

• p ∨ ∼ q

Miscellaneous Exercise 1 | Q 1.07 | Page 30

Choose the correct alternative :

The negation of the proposition “If 2 is prime, then 3 is odd”, is ______

• If 2 is not prime, then 3 is not odd.

• 2 is prime and 3 is not odd.

• 2 is not prime and 3 is odd.

• If 2 is not prime, then 3 is odd.

Miscellaneous Exercise 1 | Q 1.08 | Page 30

Choose the correct alternative :

The statement (∼ p ∧ q) ∨∼ q is

• p ∨ q

• p ∧ q

• ∼ (p ∨ q)

• ∼ (p ∧ q)

Miscellaneous Exercise 1 | Q 1.09 | Page 30

Choose the correct alternative :

Which of the following is always true?

• (p → q) ≡ ∼ q → ∼ p

• ∼ (p ∨ q) ≡ ∼ p ∨ ∼ q

• ∼ (p → q) ≡ p ∧ ∼ q

• ∼ (p ∨ q) ≡ ∼ p ∧ ∼ q

Miscellaneous Exercise 1 | Q 1.1 | Page 30

Choose the correct alternative :

∼ (p ∨ q) ∨ (∼ p ∧ q) is logically equivalent to

• ∼ p

• p

• q

• ∼ q

Miscellaneous Exercise 1 | Q 1.11 | Page 30

Choose the correct alternative :

If p and q are two statements then (p → q) ↔ (∼ q → ∼ p) is

• Tautology

• Neither (i) not (ii)

• None of the these

Miscellaneous Exercise 1 | Q 1.12 | Page 30

Choose the correct alternative :

If p is the sentence ‘This statement is false’ then

• truth value of p is T

• truth value of p is F

• p is both true and false

• p is neither true nor false

Miscellaneous Exercise 1 | Q 1.13 | Page 30

Choose the correct alternative :

Conditional p → q is equivalent to

• p → ∼ q

• ∼ p ∨ q

• ∼ p → ∼ q

• p ∨∼q

Miscellaneous Exercise 1 | Q 1.14 | Page 30

Choose the correct alternative :

Negation of the statement “This is false or That is true” is

• That is true or This is false

• That is true and This is false

• That is true and That is false

• That is false and That is true

Miscellaneous Exercise 1 | Q 1.15 | Page 30

Choose the correct alternative :

If p is any statement then (p ∨ ∼ p) is a

• contingency

• tautology

• None of them

Miscellaneous Exercise 1 | Q 2.1 | Page 30

Fill in the blanks :

The statement q → p is called as the ––––––––– of the statement p → q.

Miscellaneous Exercise 1 | Q 2.2 | Page 30

Fill in the blanks :

Conjunction of two statement p and q is symbolically written as ______.

Miscellaneous Exercise 1 | Q 2.3 | Page 30

Fill in the blanks :

If p ∨ q is true then truth value of ∼ p ∨ ∼ q is –––––––––.

Miscellaneous Exercise 1 | Q 2.4 | Page 30

Fill in the blanks :

Negation of “some men are animal” is –––––––––.

Miscellaneous Exercise 1 | Q 2.5 | Page 30

Fill in the blanks :

Truth value of if x = 2, then x2 = − 4 is –––––––––.

Miscellaneous Exercise 1 | Q 2.6 | Page 30

Fill in the blanks :

Inverse of statement pattern p ↔ q is given by –––––––––.

Miscellaneous Exercise 1 | Q 2.7 | Page 30

Fill in the blanks :

p ↔ q is false when p and q have ––––––––– truth values.

Miscellaneous Exercise 1 | Q 2.8 | Page 31

Fill in the blanks :

Let p : the problem is easy. r : It is not challenging then verbal form of ∼ p → r is –––––––––.

Miscellaneous Exercise 1 | Q 2.9 | Page 31

Fill in the blanks :

Truth value of 2 + 3 = 5 if and only if − 3 > − 9 is –––––––––.

Miscellaneous Exercise 1 | Q 3.01 | Page 31

State whether the following statement is True or False :

Truth value of 2 + 3 < 6 is F.

• True

• False

Miscellaneous Exercise 1 | Q 3.02 | Page 31

State whether the following statement is True or False :

There are 24 months in year is a statement.

• True

• False

Miscellaneous Exercise 1 | Q 3.03 | Page 31

State whether the following statement is True or False :

p ∨ q has truth value F is both p and q has truth value F.

• True

• False

Miscellaneous Exercise 1 | Q 3.04 | Page 31

State whether the following statement is True or False:

The negation of 10 + 20 = 30 is, it is false that 10 + 20 ≠ 30.

• True

• False

Miscellaneous Exercise 1 | Q 3.05 | Page 31

State whether the following statement is True or False :

Dual of (p ∧ ∼ q) ∨ t is (p ∨ ∼ q) ∨ C.

• True

• False

Miscellaneous Exercise 1 | Q 3.06 | Page 31

State whether the following statement is True or False :

Dual of “John and Ayub went to the forest” is “John and Ayub went to the forest”.

• True

• False

Miscellaneous Exercise 1 | Q 3.07 | Page 31

State whether the following statement is True or False :

“His birthday is on 29th February” is not a statement.

• True

• False

Miscellaneous Exercise 1 | Q 3.08 | Page 31

State whether the following statement is True or False :

x2 = 25 is true statement.

• True

• False

Miscellaneous Exercise 1 | Q 3.09 | Page 31

State whether the following statement is True or False :

Truth value of sqrt(5) is not an irrational number is T.

• True

• False

Miscellaneous Exercise 1 | Q 3.1 | Page 31

State whether the following statement is True or False :

p ∧ t = p.

• True

• False

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
Ice cream Sundaes are my favourite.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
x + 3 = 8 ; x is variable.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
z is a positive number.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
(a + b)2 = a2 + 2ab + b2 for all a, b ∈ R.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
(2 + 1)2 = 9.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
How beautiful the flower is!

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
The square of any odd number is even.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
All integers are natural numbers.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
If x is real number then x2 ≥ 0.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
Do not come inside the room.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.01 | Page 31

Solve the following :

State which of the following sentences are statements in logic.
What a horrible sight it was!

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

What is happy ending?

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

The square of every real number is positive.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

Every parallelogram is a rhombus.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

a2 − b2 = (a + b) (a − b) for all a, b ∈ R.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

The Himalayas is the highest mountain range.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

(x − 2) (x − 3) = x2 − 5x + 6 for all x∈R.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

What are the causes of rural unemployment?

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

0! = 1

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.02 | Page 31

Which of the following sentence is a statement? In case of a statement, write down the truth value.

The quadratic equation ax2 + bx + c = 0 (a ≠ 0) always has two real roots.

• Is a statement

• Is not a statement

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

The Sun has set and Moon has risen.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

Mona likes Mathematics and Physics.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

3 is prime number if 3 is perfect square number.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

Kavita is brilliant and brave.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

If Kiran drives the car, then Sameer will walk.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

The necessary condition for existence of a tangent to the curve of the function is continuity.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

To be brave is necessary and sufficient condition to climb the Mount Everest.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

x3 + y3 = (x + y)3 if xy = 0.

Miscellaneous Exercise 1 | Q 4.03 | Page 31

Assuming the first statement p and second as q. Write the following statement in symbolic form.

The drug is effective though it has side effects.

Miscellaneous Exercise 1 | Q 4.03 | Page 32

Assuming the first statement p and second as q. Write the following statement in symbolic form.

If a real number is not rational, then it must be irrational.

Miscellaneous Exercise 1 | Q 4.03 | Page 32

Assuming the first statement p and second as q. Write the following statement in symbolic form.

It is not true that Ram is tall and handsome.

Miscellaneous Exercise 1 | Q 4.03 | Page 32

Assuming the first statement p and second as q. Write the following statement in symbolic form.

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

Miscellaneous Exercise 1 | Q 4.03 | Page 32

Assuming the first statement p and second as q. Write the following statement in symbolic form.

It is not true that intelligent persons are neither polite nor helpful.

Miscellaneous Exercise 1 | Q 4.03 | Page 32

Assuming the first statement p and second as q. Write the following statement in symbolic form.

If the question paper is not easy then we shall not pass.

Miscellaneous Exercise 1 | Q 4.04 | Page 32

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

Proof is lengthy and it is not interesting.

Miscellaneous Exercise 1 | Q 4.04 | Page 32

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

If proof is lengthy then it is interesting.

Miscellaneous Exercise 1 | Q 4.04 | Page 32

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

It is not true that the proof is lengthy but it is interesting.

Miscellaneous Exercise 1 | Q 4.04 | Page 32

If p : Proof is lengthy.
q : It is interesting.
Express the following statement in symbolic form.

It is interesting iff the proof is lengthy.

Miscellaneous Exercise 1 | Q 4.05 | Page 32

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r :
Sachin is happy.
Write the verbal statement of the following.

(p ∧ q) ∨ r

Miscellaneous Exercise 1 | Q 4.05 | Page 32

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

p → r

Miscellaneous Exercise 1 | Q 4.05 | Page 32

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

∼ p ∨ q

Miscellaneous Exercise 1 | Q 4.05 | Page 32

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

p → (p ∧ r)

Miscellaneous Exercise 1 | Q 4.05 | Page 32

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

p → q

Miscellaneous Exercise 1 | Q 4.05 | Page 32

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

(p ∧ q) ∧ ∼ r

Miscellaneous Exercise 1 | Q 4.05 | Page 32

Let p : Sachin wins the match.
q : Sachin is a member of Rajya Sabha.
r : Sachin is happy.
Write the verbal statement of the following.

∼ (p ∨ q) ∧ r

Miscellaneous Exercise 1 | Q 4.06 | Page 32

Determine the truth value of the following statement.

4 + 5 = 7 or 9 − 2 = 5

Miscellaneous Exercise 1 | Q 4.06 | Page 32

Determine the truth value of the following statement.

If 9 > 1 then x2 − 2x + 1 = 0 for x = 1

Miscellaneous Exercise 1 | Q 4.06 | Page 32

Determine the truth value of the following statement.

x + y = 0 is the equation of a straight line if and only if y2 = 4x is the equation of the parabola.

Miscellaneous Exercise 1 | Q 4.06 | Page 32

Determine the truth value of the following statement.

It is not true that 2 + 3 = 6 or 12 + 3 =5

Miscellaneous Exercise 1 | Q 4.07 | Page 32

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

Stock prices are not high or stocks are rising.

Miscellaneous Exercise 1 | Q 4.07 | Page 32

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

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

Miscellaneous Exercise 1 | Q 4.07 | Page 32

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

If stock prices are high then stocks are not rising.

Miscellaneous Exercise 1 | Q 4.07 | Page 32

Assuming the following statement.
p : Stock prices are high.
q : Stocks are rising.
to be true, find the truth value of the following.

It is false that stocks are rising and stock prices are high.

Miscellaneous Exercise 1 | Q 4.07 | Page 32

Assuming the following statement.

p : Stock prices are high.

q : Stocks are rising.

to be true, find the truth value of the following.

Stock prices are high or stocks are not rising iff stocks are rising.

Miscellaneous Exercise 1 | Q 4.08 | Page 32

Rewrite the following statement without using conditional –
(Hint : p → q ≡ ∼ p ∨ q)

If price increases, then demand falls.

Miscellaneous Exercise 1 | Q 4.08 | Page 32

Rewrite the following statement without using conditional –
(Hint : p → q ≡ ∼ p ∨ q)

If demand falls, then price does not increase.

Miscellaneous Exercise 1 | Q 4.09 | Page 32

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

(p ∧ q) → ∼ p.

Miscellaneous Exercise 1 | Q 4.09 | Page 32

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

p ↔ (q → ∼ p)

Miscellaneous Exercise 1 | Q 4.09 | Page 32

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

(p ∧ ∼ q) ∨ (∼ p ∧ q)

Miscellaneous Exercise 1 | Q 4.09 | Page 32

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

∼ (p ∧ q) → ∼ (q ∧ p)

Miscellaneous Exercise 1 | Q 4.09 | Page 32

If p, q, r are statements with truth values T, T, F respectively determine the truth values of the following.

∼ [(p → q) ↔ (p ∧ ∼ q)]

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

If ∆ABC is not equilateral, then it is not equiangular.

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

Ramesh is intelligent and he is hard working.

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

An angle is a right angle if and only if it is of measure 90°.

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

Kanchanganga is in India and Everest is in Nepal.

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

If x ∈ A ∩ B, then x ∈ A and x ∈ B.

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∧ ~q) ↔ (q → p)

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(~p ∨ q) ∧ (~p ∧ ~q)

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∧ r) → (p ∨ ~q)

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∨ r) → ~(q ∧ r)

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∨ ~q) → (r ∧ p)

Miscellaneous Exercise 1 | Q 4.12 | Page 33

What is tautology? What is contradiction?
Show that the negation of a tautology is a contradiction and the negation of a contradiction is a tautology.

Miscellaneous Exercise 1 | Q 4.13 | Page 33

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

[(p ∧ q) ∨ (~p)] ∨ [p ∧ (~ q)]

Miscellaneous Exercise 1 | Q 4.13 | Page 33

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

[(~p ∧ q) ∧ (q ∧ r)] ∨ (~q)

Miscellaneous Exercise 1 | Q 4.13 | Page 33

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

[~(p ∨ q) → p] ↔ [(~p) ∧ (~q)]

Miscellaneous Exercise 1 | Q 4.13 | Page 33

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

[~(p ∧ q) → p] ↔ [(~p) ∧ (~q)]

Miscellaneous Exercise 1 | Q 4.13 | Page 33

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

[P → (~q ∨ r)] ↔ ~[p → (q → r)]

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

[~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

p ∧ (~p ∨ q) ≡ p ∧ q

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

~p ∧ q ≡ [(p ∨ q)] ∧ ~p

Miscellaneous Exercise 1 | Q 4.15 | Page 33

Write the converse, inverse, contrapositive of the following statement.

If 2 + 5 = 10, then 4 + 10 = 20.

Miscellaneous Exercise 1 | Q 4.15 | Page 33

Write the converse, inverse, contrapositive of the following statement.

If a man is bachelor, then he is happy.

Miscellaneous Exercise 1 | Q 4.15 | Page 33

Write the converse, inverse, contrapositive of the following statement.

If I do not work hard, then I do not prosper.

Miscellaneous Exercise 1 | Q 4.16 | Page 33

State the dual of the following statement by applying the principle of duality.

(p ∧ ~q) ∨ (~ p ∧ q) ≡ (p ∨ q) ∧ ~(p ∧ q)

Miscellaneous Exercise 1 | Q 4.16 | Page 33

State the dual of the following statement by applying the principle of duality.

p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)]

Miscellaneous Exercise 1 | Q 4.16 | Page 33

State the dual of the following statement by applying the principle of duality.

2 is even number or 9 is a perfect square.

Miscellaneous Exercise 1 | Q 4.17 | Page 33

Rewrite the following statement without using the connective ‘If ... then’.

If a quadrilateral is rhombus then it is not a square.

Miscellaneous Exercise 1 | Q 4.17 | Page 33

Rewrite the following statement without using the connective ‘If ... then’.

If 10 − 3 = 7 then 10 × 3 ≠ 30.

Miscellaneous Exercise 1 | Q 4.17 | Page 33

Rewrite the following statement without using the connective ‘If ... then’.

If it rains then the principal declares a holiday.

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

(~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q)

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (q ∨ r)

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

~(p ∨ q) ≡ ~p ∧ ~q

Miscellaneous Exercise 1 | Q 4.19 | Page 33

Consider the following statements.
i. If D is dog, then D is very good.
ii. If D is very good, then D is dog.
iii. If D is not very good, then D is not a dog.
iv. If D is not a dog, then D is not very good. Identify the pairs of statements having the same meaning. Justify.

Miscellaneous Exercise 1 | Q 4.2 | Page 33

Express the truth of the following statement by the Venn diagram.

All men are mortal.

Miscellaneous Exercise 1 | Q 4.2 | Page 33

Express the truth of the following statement by the Venn diagram.

Some persons are not politician.

Miscellaneous Exercise 1 | Q 4.2 | Page 33

Express the truth of the following statement by the Venn diagram.

Some members of the present Indian cricket are not committed.

Miscellaneous Exercise 1 | Q 4.2 | Page 33

Express the truth of the following statement by the Venn diagram.

Miscellaneous Exercise 1 | Q 4.21 | Page 34

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∃ x ∈ A, such that 3x + 2 > 9

Miscellaneous Exercise 1 | Q 4.21 | Page 34

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∀ x ∈ A, x2 < 18.

Miscellaneous Exercise 1 | Q 4.21 | Page 34

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∃ x ∈ A, such that x + 3 < 11.

Miscellaneous Exercise 1 | Q 4.21 | Page 34

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of the following statement.

∀ x ∈ A, x2 + 2 ≥ 5.

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

7 is prime number and Tajmahal is in Agra.

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

10 > 5 and 3 < 8

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

I will have tea or coffee.

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

∀ n ∈ N, n + 3 > 9.

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

∃ n ∈ A, such that x + 5 < 11.

## Chapter 1: Mathematical Logic

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Exercise 1.6Exercise 1.7Exercise 1.8Exercise 1.9Exercise 1.10Miscellaneous Exercise 1

## Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 1 - Mathematical Logic

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