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## Chapter 1: Mathematical Logic

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

**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

**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°

**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!

Please grant me a loan.

`sqrt(-4)` is an irrational number.

x^{2} − 6x + 8 = 0 implies x = −4 or x = −2.

He is an actor.

Did you eat lunch yet?

Have a cup of cappuccino.

(x + y)^{2} = x^{2} + 2xy + y^{2} for all x, y ∈ R.

Every real number is a complex number.

1 is a prime number.

With the sunset the day ends.

1 ! = 0

3 + 5 > 11

The number π is an irrational number.

x^{2} - y^{2} = (x + y)(x - y) for all x, y ∈ R.

The number 2 is the only even prime number.

Two co-planar lines are either parallel or intersecting.

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

Give me a compass box.

Bring the motor car here.

It may rain today.

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

Can you speak in English?

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

**Express the following statement in symbolic form.**

e is a vowel or 2 + 3 = 5

**Express the following statement in symbolic form.**

Mango is a fruit but potato is a vegetable.

**Express the following statement in symbolic form.**

Milk is white or grass is green.

**Express the following statement in symbolic form.**

I like playing but not singing.

**Express the following statement in symbolic form.**

Even though it is cloudy, it is still raining.

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

Earth is a planet and Moon is a star.

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

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

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

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

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

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

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

**Write the negation of the following statement.**

All men are animals.

**Write the negation of the following statement.**

− 3 is a natural number.

**Write the negation of the following statement.**

It is false that Nagpur is capital of Maharashtra

**Write the negation of the following statement.**

2 + 3 ≠ 5

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

`sqrt5` is an irrational number.

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

London is in England.

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

For every x ∈ N, x + 3 < 8.

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

**Write the following statement in symbolic form.**

If triangle is equilateral then it is equiangular.

**Write the following statement in symbolic form.**

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

**Write the following statement in symbolic form.**

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

**Write the following statement in symbolic form.**

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

**Write the following statement in symbolic form.**

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

**Write the following statement in symbolic form.**

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

**Find the truth value of the following statement.**

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

**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.

**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.

**Find the truth value of the following statement.**

Neither 27 is a prime number nor divisible by 4.

**Find the truth value of the following statement.**

3 is a prime number and an odd number.

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

p ∧ (q ∧ r)

**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)

**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)]

**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)

**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]

**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]

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.

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.

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.

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

**p ↔ ~ q**

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

**~ (p ∨ q)**

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

**q → p**

If p : He swims

q : Water is warm

Give the verbal statement for the following symbolic statement.

**q ∧ ~ p**

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

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

x^{2} + 3x - 10 = 0

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

3x - 4 < 9

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

n^{2} ≥ 1

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

2n - 1 = 5

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

y + 4 > 6

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

3y - 2 ≤ 9

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

If B = {2, 3, 5, 6, 7} determine the truth value of

∃ n ∈ B, such that n + 6 > 12.

If B = {2, 3, 5, 6, 7} determine the truth value of

∃ n ∈ B, such that 2n + 2 < 4.

If B = {2, 3, 5, 6, 7} determine the truth value of

∀ y ∈ B, such that y^{2} is negative.

If B = {2, 3, 5, 6, 7} determine the truth value of

∀ y ∈ B, such that (y - 5) ∈ N

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

**Prepare truth tables for the following statement pattern.**

p → (~ p ∨ q)

**Prepare truth tables for the following statement pattern.**

(~ p ∨ q) ∧ (~ p ∨ ~ q)

**Prepare truth tables for the following statement pattern.**

(p ∧ r) → (p ∨ ~ q)

**Prepare truth table for (p ˄ q) ˅ ~ r**

(p ∧ q) ∨ ~ r

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

q ∨ [~ (p ∧ q)]

**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.**

(p ∧ ~ q) → (~ p ∧ ~ q)

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

~ p → (p → ~ q)

**Prove that the following statement pattern is a tautology.**

(p ∧ q) → q

**Prove that the following statement pattern is a tautology.**

(p → q) ↔ (~ q → ~ p)

**Prove that the following statement pattern is a tautology.**

(~p ∧ ~q ) → (p → q)

**Prove that the following statement pattern is a tautology.**

(~ p ∨ ~ q) ↔ ~ (p ∧ q)

**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

**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)

**Show that the following statement pattern is contingency.**

(p∧~q) → (~p∧~q)

**Show that the following statement pattern is contingency.**

(p → q) ↔ (~ p ∨ q)

**Show that the following statement pattern is contingency.**

p ∧ [(p → ~ q) → q]

**Show that the following statement pattern is contingency.**

(p → q) ∧ (p → r)

**Using the truth table, verify**

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

**Using the truth table, verify**

p → (p → q) ≡ ~ q → (p → q)

**Using the truth table, verify**

~(p → ~q) ≡ p ∧ ~ (~ q) ≡ p ∧ q

**Using the truth table, verify**

~(p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

**Using the truth table, verify**

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

**Prove that the following pair of statement pattern is equivalent.**

p ↔ q and (p → q) ∧ (q → p)

**Prove that the following pair of statement pattern is equivalent.**

p → q and ~ q → ~ p and ~ p ∨ q

**Prove that the following pair of statement pattern is equivalent.**

~(p ∧ q) and ~p ∨ ~q

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

**Write the dual of the following:**

(p ∨ q) ∨ r

**Write the dual of the following:**

~(p ∨ q) ∧ [p ∨ ~ (q ∧ ~ r)]

**Write the dual of the following:**

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r

**Write the dual of the following:**

~(p ∧ q) ≡ ~ p ∨ ~ q

**Write the dual statement of the following compound statement.**

13 is prime number and India is a democratic country.

**Write the dual statement of the following compound statement.**

Karina is very good or everybody likes her.

**Write the dual statement of the following compound statement.**

Radha and Sushmita cannot read Urdu.

**Write the dual statement of the following compound statement.**

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

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

**Write the negation of the following statement.**

All the stars are shining if it is night.

**Write the negation of the following statement.**

∀ n ∈ N, n + 1 > 0

**Write the negation of the following statement.**

∃ n ∈ N, (n^{2} + 2) is odd number.

**Write the negation of the following statement.**

Some continuous functions are differentiable.

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

(p → r) ∧ q

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

~(p ∨ q) → r

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

(~p ∧ q) ∧ (~q ∨ ~r)

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

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

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

If he studies, then he will go to college.

**With proper justification, state the negation of the following.**

(p → q) ∨ (p → r)

**With proper justification, state the negation of the following.**

(p ↔ q) v (~ q → ~r)

**With proper justification, state the negation of the following.**

(p → q) ∧ r

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

**Without using truth table, show that**

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

**Without using truth table, show that**

p ∧ [(~ p ∨ q) ∨ ~ q] ≡ p

**Without using truth table, show that**

~ [(p ∧ q) → ~ q] ≡ p ∧ q

**Without using truth table, show that**

~r → ~ (p ∧ q) ≡ [~ (q → r)] → ~ p

**Without using truth table, show that**

(p ∨ q) → r ≡ (p → r) ∧ (q → r)

**Using the algebra of statement, prove that**

[p ∧ (q ∨ r)] ∨ [~ r ∧ ~ q ∧ p] ≡ p

**Using the algebra of statement, prove that**

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

**Using the algebra of statement, prove that**

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

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 1 Mathematical LogicExercise 1.10 [Pages 22 - 27]

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

Some hardworking students are obedient.

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

No circles are polygons.

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

All teachers are scholars and scholars are teachers.

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

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

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

Some share brokers are chartered accountants.

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

No wicket keeper is bowler, in a cricket team.

**Represent the following statement by the Venn diagram.**

Some non-resident Indians are not rich.

**Represent the following statement by the Venn diagram.**

No circle is rectangle.

**Represent the following statement by the Venn diagram.**

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

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

**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

**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.

**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

**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

**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.

**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

**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.

**Choose the correct alternative :**

The statement (∼ p ∧ q) ∨∼ q is

p ∨ q

p ∧ q

∼ (p ∨ q)

∼ (p ∧ q)

**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

**Choose the correct alternative :**

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

∼ p

p

q

∼ q

**Choose the correct alternative :**

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

Contradiction

Tautology

Neither (i) not (ii)

None of the these

**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

**Choose the correct alternative :**

Conditional p → q is equivalent to

p → ∼ q

∼ p ∨ q

∼ p → ∼ q

p ∨∼q

**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

**Choose the correct alternative :**

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

contingency

contradiction

tautology

None of them

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**Fill in the blanks :**

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

**State whether the following statement is True or False :**

Truth value of 2 + 3 < 6 is F.

True

False

**State whether the following statement is True or False :**

There are 24 months in year is a statement.

True

False

**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

**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

**State whether the following statement is True or False :**

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

True

False

**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

**State whether the following statement is True or False :**

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

True

False

**State whether the following statement is True or False :**

x^{2} = 25 is true statement.

True

False

**State whether the following statement is True or False :**

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

True

False

**State whether the following statement is True or False :**

p ∧ t = p.

True

False

**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

**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

**Solve the following :**

State which of the following sentences are statements in logic.

Read a lot to improve your writing skill.

Is a statement

Is not a statement

**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

**Solve the following :**

State which of the following sentences are statements in logic.

(a + b)^{2} = a^{2} + 2ab + b^{2} for all a, b ∈ R.

Is a statement

Is not a statement

**Solve the following :**

State which of the following sentences are statements in logic.

(2 + 1)^{2} = 9.

Is a statement

Is not a statement

**Solve the following :**

State which of the following sentences are statements in logic.

Why are you sad?

Is a statement

Is not a statement

**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

**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

**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

**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

**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

**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

**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

**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

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

a^{2} − b^{2} = (a + b) (a − b) for all a, b ∈ R.

Is a statement

Is not a statement

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

Please carry out my instruction.

Is a statement

Is not a statement

**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

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

(x − 2) (x − 3) = x^{2} − 5x + 6 for all x∈R.

Is a statement

Is not a statement

**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

**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

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

The quadratic equation ax^{2} + bx + c = 0 (a ≠ 0) always has two real roots.

Is a statement

Is not a statement

**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

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

The Sun has set and Moon has risen.

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

Mona likes Mathematics and Physics.

**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.

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

Kavita is brilliant and brave.

**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.

**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.

**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.

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

x^{3} + y^{3} = (x + y)^{3} if xy = 0.

**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.

**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.

**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.

**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.

**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.

**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.

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

Proof is lengthy and it is not interesting.

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

If proof is lengthy then it is interesting.

**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.

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

It is interesting iff the proof is lengthy.

**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.

Write the verbal statement of the following.

(p ∧ q) ∨ r

**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

**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

**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)

**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

**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

**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

**Determine the truth value of the following statement.**

4 + 5 = 7 or 9 − 2 = 5

**Determine the truth value of the following statement.**

If 9 > 1 then x^{2} − 2x + 1 = 0 for x = 1

**Determine the truth value of the following statement.**

x + y = 0 is the equation of a straight line if and only if y^{2} = 4x is the equation of the parabola.

**Determine the truth value of the following statement.**

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

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.**

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.**

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.**

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.**

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.**

**Rewrite the following statement without using conditional –**

(Hint : p → q ≡ ∼ p ∨ q)

If price increases, then demand falls.

**Rewrite the following statement without using conditional – **

(Hint : p → q ≡ ∼ p ∨ q)

If demand falls, then price does not increase.

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

(p ∧ q) → ∼ p.

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

p ↔ (q → ∼ p)

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

(p ∧ ∼ q) ∨ (∼ p ∧ q)

∼ (p ∧ q) → ∼ (q ∧ p)

∼ [(p → q) ↔ (p ∧ ∼ q)]

**Write the negation of the following.**

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

**Write the negation of the following.**

Ramesh is intelligent and he is hard working.

**Write the negation of the following.**

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

**Write the negation of the following.**

Kanchanganga is in India and Everest is in Nepal.

**Write the negation of the following.**

If x ∈ A ∩ B, then x ∈ A and x ∈ B.

**Construct the truth table for the following statement pattern.**

(p ∧ ~q) ↔ (q → p)

**Construct the truth table for the following statement pattern.**

(~p ∨ q) ∧ (~p ∧ ~q)

**Construct the truth table for the following statement pattern.**

(p ∧ r) → (p ∨ ~q)

**Construct the truth table for the following statement pattern.**

(p ∨ r) → ~(q ∧ r)

**Construct the truth table for the following statement pattern.**

(p ∨ ~q) → (r ∧ p)

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.

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

[(p ∧ q) ∨ (~p)] ∨ [p ∧ (~ q)]

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

[(~p ∧ q) ∧ (q ∧ r)] ∨ (~q)

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

[~(p ∨ q) → p] ↔ [(~p) ∧ (~q)]

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

[~(p ∧ q) → p] ↔ [(~p) ∧ (~q)]

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

[P → (~q ∨ r)] ↔ ~[p → (q → r)]

**Using the truth table, prove the following logical equivalence.**

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

**Using the truth table, prove the following logical equivalence.**

[~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

**Using the truth table, prove the following logical equivalence.**

p ∧ (~p ∨ q) ≡ p ∧ q

**Using the truth table, prove the following logical equivalence.**

p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

**Using the truth table, prove the following logical equivalence.**

~p ∧ q ≡ [(p ∨ q)] ∧ ~p

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

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

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

If a man is bachelor, then he is happy.

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

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

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

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

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

p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)]

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

2 is even number or 9 is a perfect square.

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

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

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

If 10 − 3 = 7 then 10 × 3 ≠ 30.

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

If it rains then the principal declares a holiday.

**Write the dual of the following.**

(~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q)

**Write the dual of the following.**

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

**Write the dual of the following.**

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (q ∨ r)

**Write the dual of the following.**

~(p ∨ q) ≡ ~p ∧ ~q

**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.

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

All men are mortal.

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

Some persons are not politician.

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

Some members of the present Indian cricket are not committed.

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

No child is an adult.

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

∃ x ∈ A, such that 3x + 2 > 9

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

∀ x ∈ A, x^{2} < 18.

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

∃ x ∈ A, such that x + 3 < 11.

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

∀ x ∈ A, x^{2} + 2 ≥ 5.

**Write the negation of the following statement.**

7 is prime number and Tajmahal is in Agra.

**Write the negation of the following statement.**

10 > 5 and 3 < 8

**Write the negation of the following statement.**

I will have tea or coffee.

**Write the negation of the following statement.**

∀ n ∈ N, n + 3 > 9.

**Write the negation of the following statement.**

∃ n ∈ A, such that x + 5 < 11.

## Chapter 1: Mathematical Logic

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

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 1 (Mathematical Logic) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 1 Mathematical Logic are Truth Value of Statement, Quantifier and Quantified Statements in Logic, Statement Patterns and Logical Equivalence, Algebra of Statements, Venn Diagrams, Logical Connective, Simple and Compound Statements.

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