SSC BOARD PAPERS IMPORTANT TOPICS COVERED FOR BOARD EXAM 2024

### Miscellaneous Exercise 1 [Pages 29 - 34] Chapter 1 Mathematical Logic Commerce

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]

#### QUESTION

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

• Come here

#### QUESTION

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.

#### SOLUTION

x is a natural number.

#### QUESTION

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

#### SOLUTION

distributive law.

#### QUESTION

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

#### SOLUTION

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

#### QUESTION

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.

#### SOLUTION

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

#### QUESTION

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

∼ (p ∨ q)

#### QUESTION

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.

#### SOLUTION

2 is prime and 3 is not odd.

#### QUESTION

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)

∼ (p ∧ q).

#### QUESTION

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

#### SOLUTION

∼ (p → q) ≡ p ∧ ∼ q.

#### QUESTION

Miscellaneous Exercise 1 | Q 1.1 | Page 30

Choose the correct alternative :

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

• ∼ p

• p

• q

• ∼ q

∼ p.

#### QUESTION

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

tautology.

#### QUESTION

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

#### SOLUTION

p is neither true nor false

#### QUESTION

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

∼p ∨ q.

#### QUESTION

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

#### SOLUTION

That is true and That is false.

#### QUESTION

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

tautology.

#### QUESTION

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.

#### SOLUTION

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

#### QUESTION

Miscellaneous Exercise 1 | Q 2.2 | Page 30

Fill in the blanks :

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

#### QUESTION

Miscellaneous Exercise 1 | Q 2.3 | Page 30

Fill in the blanks :

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

#### SOLUTION

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

#### QUESTION

Miscellaneous Exercise 1 | Q 2.4 | Page 30

Fill in the blanks :

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

#### SOLUTION

Negation of “some men are animal” is No men are animals.

#### QUESTION

Miscellaneous Exercise 1 | Q 2.5 | Page 30

Fill in the blanks :

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

#### SOLUTION

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

#### QUESTION

Miscellaneous Exercise 1 | Q 2.6 | Page 30

Fill in the blanks :

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

#### SOLUTION

Inverse of statement pattern p ↔ q is given by ∼ p → ∼ q.

#### QUESTION

Miscellaneous Exercise 1 | Q 2.7 | Page 30

Fill in the blanks :

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

#### SOLUTION

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

#### QUESTION

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 –––––––––.

#### SOLUTION

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

#### QUESTION

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 –––––––––.

#### SOLUTION

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

#### QUESTION

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

False.

#### QUESTION

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

True.

#### QUESTION

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

False.

#### QUESTION

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

False.

#### QUESTION

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

False.

#### QUESTION

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

True.

#### QUESTION

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

True.

#### QUESTION

Miscellaneous Exercise 1 | Q 3.08 | Page 31

State whether the following statement is True or False :

x2 = 25 is true statement.

• True

• False

False.

#### QUESTION

Miscellaneous Exercise 1 | Q 3.09 | Page 31

State whether the following statement is True or False :

Truth value of $5$ √5 is not an irrational number is T.

• True

• False

False.

#### QUESTION

Miscellaneous Exercise 1 | Q 3.1 | Page 31

State whether the following statement is True or False :

p ∧ t = p.

• True

• False

True.

#### QUESTION

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

Is a statement

#### QUESTION

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

Is a statement

#### QUESTION

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

#### SOLUTION

Is not a statement

#### QUESTION

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

Is a statement

#### QUESTION

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

Is a statement

#### QUESTION

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

Is a statement

#### QUESTION

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

#### SOLUTION

Is not a statement

#### QUESTION

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

#### SOLUTION

Is not a statement

#### QUESTION

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

Is a statement

#### QUESTION

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

Is a statement

#### QUESTION

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

Is a statement

#### QUESTION

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

#### SOLUTION

Is not a statement

#### QUESTION

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

#### SOLUTION

Is not a statement

#### QUESTION

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

#### SOLUTION

It is a statement which is false. Hence, its truth value is F.

#### QUESTION

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

#### SOLUTION

It is a statement which is false. Hence, its truth value is F.

#### QUESTION

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

#### SOLUTION

It is a statement which is true. Hence, its truth value is T.

#### QUESTION

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

#### SOLUTION

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

#### QUESTION

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

#### SOLUTION

It is a statement which is true. Hence, its truth value is T.

#### QUESTION

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

#### SOLUTION

It is a statement which is true. Hence, its truth value is T.

#### QUESTION

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

#### SOLUTION

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

#### QUESTION

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

#### SOLUTION

It is a statement which is true. Hence, its truth value is T.

#### QUESTION

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

#### SOLUTION

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

Hence, its truth value is F.

#### QUESTION

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

#### SOLUTION

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

#### QUESTION

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.

#### SOLUTION

Let p : The sun has set.
q : The moon has risen

The symbolic form is p ∧ q.

#### QUESTION

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.

#### SOLUTION

Let p : Mona likes Mathematics
q : Mona likes Physics

The symbolic form is p ∧ q.

#### QUESTION

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.

#### SOLUTION

Let p : 3 is a prime number.
q : 3 is a perfect square number.

The symbolic form is p ↔ q.

#### QUESTION

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.

#### SOLUTION

Let p : Kavita is brilliant.
q : Kavita is brave.

The symbolic form is p ∧ q.

#### QUESTION

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.

#### SOLUTION

Let p : Kiran drives the car.

q : Sameer will walk.

The symbolic form is p → q.

#### QUESTION

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.

#### SOLUTION

The given statement can also be expressed as ‘If the function is continuous, then the tangent to the curve exists’.

Let p : The function is continuous
q : The tangent to the curve exists.

∴ p → q is the symbolic form of the given statement.

#### QUESTION

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.

#### SOLUTION

Let p : To be brave

q : climb the Mount Everest

∴ p ↔ q is the symbolic form of the given statement.

#### QUESTION

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.

#### SOLUTION

Let p : x3 + y3 = (x + y)3

q : xy = 0

∴ p ↔ q is the symbolic form of the given statement.

#### QUESTION

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.

#### SOLUTION

The given statement can also be expressed as “The drug is effective and it has side effects”

Let p : The drug is effective.
q : It has side effects.

∴ p ∧ q is the symbolic form of the given statement.

#### QUESTION

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.

#### SOLUTION

Let p : A real number is not rational.
q : A real number must be irrational.

The symbolic form is p → q.

#### QUESTION

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.

#### SOLUTION

Let p : Ram is tall.
q : Ram is handsome.

The symbolic form is ∼(p ∧ q).

#### QUESTION

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.

#### SOLUTION

Let p : it is cloudy.
q : It is still raining.

The symbolic form is ~ p ∧ q.

#### QUESTION

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.

#### SOLUTION

Let p : Intelligent persons are neither polite nor helpful

The symbolic form is ∼ p.

Alternate method:

Let p : Intelligent persons are polite.
q : Intelligent persons are helpful.
The symbolic form is ~(~ p ∧ ~ q).

#### QUESTION

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.

#### SOLUTION

Let p : The question paper is not easy.
q : We shall not pass.

The symbolic form is p → q.

#### QUESTION

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.

p ∧ ∼ q

#### QUESTION

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.

p → q

#### QUESTION

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.

∼(p ∧ q)

#### QUESTION

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.

q ↔ p

#### QUESTION

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

#### SOLUTION

Sachin wins the match or he is the member of Rajya Sabha or Sachin is happy.

#### QUESTION

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

#### SOLUTION

If Sachin wins the match then he is happy.

#### QUESTION

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

#### SOLUTION

Sachin does not win the match or he is the member of Rajya Sabha.

#### QUESTION

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)

#### SOLUTION

If sachin wins the match, then he is the member of Rajyasabha or he is happy.

#### QUESTION

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

#### SOLUTION

If Sachin wins the match then he is a member of Rajyasabha.

#### QUESTION

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

#### SOLUTION

Sachin wins the match and he is the member of Rajyasabha but he is not happy.

#### QUESTION

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

#### SOLUTION

It is false that Sachin wins the match or he is the member of Rajyasabha but he is happy.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.06 | Page 32

Determine the truth value of the following statement.

4 + 5 = 7 or 9 − 2 = 5

#### SOLUTION

Let p : 4 + 5 = 7
q : 9 – 2 = 5
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.

#### QUESTION

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

#### SOLUTION

Let p : 9 > 1
q : x2 – 2x + 1 = 0 for x = 1

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.

#### QUESTION

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.

#### SOLUTION

Let p : x + y = 0 is the equation of a straight line.
q : y2 = 4x is the equation of the parabola.

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.

#### QUESTION

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

#### SOLUTION

Let p : 2 + 3 = 6
q : 12 + 3 = 5
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 ≡ T

∴ Truth value of the given statement is T.

#### QUESTION

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.

#### SOLUTION

Given that the truth values of both p and q are T.

The symbolic form of the given statement is ~ p ∨ q.

∴ ~ p ∨ q ≡ ~ T ∨ T ≡ F ∨ T

Hence, truth value is T.

#### QUESTION

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.

#### SOLUTION

The symbolic form of the given statement is
(p ∧ q) ↔ p.

∴ (p ∧ q) ↔ p ≡ (T ∧ T) ↔ T

≡ T ↔ T

≡ T

Hence, truth value is T.

#### QUESTION

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.

#### SOLUTION

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

∴ p → ~ q ≡ T → ~ T ≡ T → F ≡ F

Hence, truth value is F.

#### QUESTION

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.

#### SOLUTION

The symbolic form of the given statement is ~(q ∧ p).

∴ ~(q ∧ p) ≡ ~(T ∧ T) ≡ ~T ≡ F

Hence, truth value is F.

#### QUESTION

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.

#### SOLUTION

The symbolic form of the given statement is (p ∨ ~q) ↔ q.

∴ (p ∨ ~q) ↔ q ≡ (T ∨ ~T) ↔ T

≡ (T ∨ F) ↔ T

≡ T ↔ T

≡ T

Hence, truth value is T.

#### QUESTION

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.

#### SOLUTION

Let p : Prince increases.
q : demand falls.
The given statement is p → q.
But p → q ≡ ~p ∨ q.

The given statement can be written as ‘Price does not increase or demand falls’.

#### QUESTION

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.

#### SOLUTION

Let p : demand falls.
q : Price does not increase.
The given statement is p → q.

But p → q ≡ ~ p ∨ q.

∴ The given statement can be written as ‘Demand does not fall or price does not increase’.

#### QUESTION

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.

#### SOLUTION

(p ∧ q) → ∼ p ≡ (T ∧ T) → ∼ T

≡ T  → F

≡ F.

Hence, truth value is F.

#### QUESTION

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)

#### SOLUTION

p ↔ (q → ∼ p) ≡ T ↔ (T → ∼ T)

≡ T ↔ (T → F)

≡ T ↔ F

≡ F

Hence, truth value is F.

#### QUESTION

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)

#### SOLUTION

(p ∧ ∼ q) ∨ (∼ p ∧ q) ≡ (T ∧ ∼ T) ∨ (∼ T ∧ T)

≡ (T ∧ F) ∨ (F ∧ T)

≡ F ∨ F

≡ F

Hence, truth value is F.

#### QUESTION

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)

#### SOLUTION

∼ (p ∧ q) → ∼ (q ∧ p) ≡ ∼ (T ∧ T) → ∼ (T ∧ T)

≡ ~ T → ~ T

≡ F → F

≡ T

Hence, truth value is T.

#### QUESTION

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)]

#### SOLUTION

∼[(p → q) ↔ (p ∧ ∼q)] ≡ ∼ [(T → T) ↔ (T ∧ ∼ T)]

≡ ~[T ↔ (T ∧ F)]

≡ ~(T ↔ F)

≡ ~ F

≡ T

Hence, truth value is T.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

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

#### SOLUTION

Let p : ∆ ABC is not equilateral.
q : ∆ ABC is not equiangular.

The given statement is p → q.

Its negation is ~(p → q) ≡ p ∧ ~ q

∴ The negation of given statement is '∆ ABC is not equilateral and it is equiangular'.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

Ramesh is intelligent and he is hard working.

#### SOLUTION

Let p : Ramesh is intelligent.
q : Ramesh is hard working.
The given statement is p ∧ q.

Its negation is ~(p ∧ q) ≡ ~ p ∨ ~ q

∴ The negation of the given statement is ‘Ramesh is not intelligent or he is not hard-working.’

#### QUESTION

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°.

#### SOLUTION

Let p : An angle is a right angle.
q : An angle is of measure 90°.

The given statement is p ↔ q.

Its negation is ~(p ↔ q) ≡ (p ∧ ~ q) ∨ (q ∧ ~ p)

∴ The negation of the given statement is ‘An angle is a right angle and it is not of measure 90° or an angle is of measure 90° and it is not a right angle.’

#### QUESTION

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

Kanchanganga is in India and Everest is in Nepal.

#### SOLUTION

Let p : Kanchanganga is in India.
q : Everest is in Nepal.
The given statement is p ∧ q.

Its negation is ~(p ∧ q) ≡ ~ p ∨ ~ q.

The negation of a given statement is ‘Kanchanganga is not in India or Everest is not in Nepal’.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

If x ∈ A ∩ B, then x ∈ A and x ∈ B.

#### SOLUTION

Let p : x ∈ A ∩ B

q : x ∈ A

r : x ∈ B

The given statement is p → (q ∧ r).

Its negation is ~[p → (q ∧ r)], and

~[p → (q ∧ r)] ≡ p ∧ ~ (q ∧ r) ≡ p ∧ ~ q ∨ ~ r

∴ The negation of given statement is x ∈ A ∩ B and x ∉ A or x ∉ B.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∧ ~q) ↔ (q → p)

#### SOLUTION

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

#### QUESTION

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(~p ∨ q) ∧ (~p ∧ ~q)

#### SOLUTION

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

#### QUESTION

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∧ r) → (p ∨ ~q)

#### SOLUTION

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

#### QUESTION

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∨ r) → ~(q ∧ r)

#### SOLUTION

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

#### QUESTION

Miscellaneous Exercise 1 | Q 4.11 | Page 33

Construct the truth table for the following statement pattern.

(p ∨ ~q) → (r ∧ p)

#### SOLUTION

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

#### QUESTION

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.

#### SOLUTION

• Tautology:
A statement pattern having truth value always T, irrespective of the truth values of its component statement is called a tautology.
A statement pattern having truth value always F, irrespective of the truth values of its component statement is called a contradiction.

Let Statement p tautology. Consider, truth table

 p ~ p T F

i.e., negation of tautology is contradiction.
Let statement of contradiction. Consider, truth table

 q ~ q F T

i.e., negation of contradiction is tautology.

#### QUESTION

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)]

#### SOLUTION

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

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

#### QUESTION

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)

#### SOLUTION

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

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

#### QUESTION

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)]

#### SOLUTION

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

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

#### QUESTION

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)]

#### SOLUTION

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

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

#### QUESTION

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)]

#### SOLUTION

 p q r ~q ~q∨r q→r p→(q→r) P→(~q∨r) ~[p→(q→r)] [P→(~q∨r)]↔~[p → (q → r)] T T T F T T T T F F T T F F F F F F T F T F T T T T T T F F T F F T T T T T F F F T T F T T T T F F F T F F F F T T F F F F T T T T T T F F F F F T T T T T F F

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

#### QUESTION

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

#### SOLUTION

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

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

∴ p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

#### QUESTION

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

[~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

#### SOLUTION

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

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

∴ [~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

#### QUESTION

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

p ∧ (~p ∨ q) ≡ p ∧ q

#### SOLUTION

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

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

∴ p ∧ (~p ∨ q) ≡ p ∧ q

#### QUESTION

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

#### SOLUTION

 1 2 3 4 5 6 7 8 9 10 p q ~p ~q p↔q p∧~q ~(p∧~q) (q∧~p) ~(q∧~p) ~(p∧~q)∧~(q ∧ ~p) T T F F T F T F T T T F F T F T F F T F F T T F F F T T F F F F T T T F T F T T

In the above truth table, the entries in columns 5 and 10 are identical.

∴ p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

#### QUESTION

Miscellaneous Exercise 1 | Q 4.14 | Page 33

Using the truth table, prove the following logical equivalence.

~p ∧ q ≡ [(p ∨ q)] ∧ ~p

#### SOLUTION

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

In the above truth table, the entries in columns 4 and 6 are identical.

∴ ~p ∧ q ≡ [(p ∨ q)] ∧ ~p

#### QUESTION

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.

#### SOLUTION

Let p : 2 + 5 = 10
q : 4 + 10 = 20

∴ The given statement is p → q.

Its converse is q → p.
If 4 + 10 = 20, then 2 + 5 = 10

Its inverse is ~p → ~q.
If 2 + 5 ≠ 10 then 4 + 10 ≠ 20.

Its contrapositive is ~q → ~p.
If 4 + 10 ≠ 20 then 2 + 5 ≠ 10.

#### QUESTION

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.

#### SOLUTION

Let p : A man is bachelor.
q : A man is happy.

∴ The given statement is p → q.

Its converse is q → p.
If a man is happy then he is bachelor.

Its inverse is ~p → ~q.
If a man is not bachelor then he is not happy.

Its contrapositive is ~q → ~p.
If a man is not happy then he is not bachelor.

#### QUESTION

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.

#### SOLUTION

Let p : I do not work hard.
q : I do not prosper.

∴ The given statement is p → q.

Its converse is q → p.
If I do not prosper then I do not work hard.

Its inverse is ~p → ~q.
If I work hard then I prosper.

Its contrapositive is ~q → ~p.
If I prosper then I work hard.

#### QUESTION

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)

#### SOLUTION

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

#### QUESTION

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)]

#### SOLUTION

p ∧ (q ∧ r) ≡ ~[(p ∨ q) ∧ (r ∧ s)]

#### QUESTION

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.

#### SOLUTION

2 is even number and 9 is a perfect square.

#### QUESTION

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.

#### SOLUTION

Let p : A quadrilateral is rhombus.

q : A quadrilateral is not a square.

The given statement is p → q.

But p → q ≡ ~p ∨ q.

∴ The given statement can be written as ‘A quadrilateral is not a rhombus or it is not a square’.

#### QUESTION

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.

#### SOLUTION

Let p : 10 − 3 = 7

q : 10 × 3 ≠ 30

The given statement is p → q.

But p → q ≡ ~p ∨ q.

∴ The given statement can be written as
'10 - 3 ≠ 7 or 10 × 3 ≠ 30'.

#### QUESTION

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.

#### SOLUTION

Let p : It rains.

q : The principal declares a holiday.

The given statement is p → q.

But p → q ≡ ~p ∨ q.

∴ The given statement can be written as ‘It does not rain or the principal declares a holiday’.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

(~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q)

#### SOLUTION

(~p ∨ q) ∧ (p ∨ ~q) ∧ (~p ∨ ~q)

#### QUESTION

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

#### SOLUTION

(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

#### QUESTION

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (q ∨ r)

#### SOLUTION

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (q ∧ r)

#### QUESTION

Miscellaneous Exercise 1 | Q 4.18 | Page 33

Write the dual of the following.

~(p ∨ q) ≡ ~p ∧ ~q

#### SOLUTION

~(p ∧ q) ≡ ~p ∨ ~q

#### QUESTION

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.

#### SOLUTION

Let p : D is dog.
q : D is very good.
Then the given statement in the symbolic form is i. p → q
ii. q → p
iii. ~q → ~p
iv. ~p → ~q

Since a statement and its contrapositive are equivalent, statements (i) and (iii) have the same meaning.
Since converse and inverse of a compound statement are equivalent, statements (ii) and (iv) have same meaning.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.2 | Page 33

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

All men are mortal.

#### SOLUTION

U : The set of all human being
A : The set of all men
B : The set of all mortal

The above Venn diagram represents the truth of the given statement, i.e. A ⊂ B.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.2 | Page 33

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

Some persons are not politician.

#### SOLUTION

U : The set of all human beings.
X : The set of all persons.
Y : The set of all politician

The above Venn diagram represents the truth of the given statement, i.e. Y - X ≠ Φ

#### QUESTION

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.

#### SOLUTION

U : The set of all human beings.
M : The set of all members of the present Indian cricket.
C : The set of all committed members of the present Indian cricket.

The above Venn diagram represents the truth of the given statement, i.e. C - M = Φ

#### QUESTION

Miscellaneous Exercise 1 | Q 4.2 | Page 33

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

#### SOLUTION

U : Set of all human beings.
C : Set of all child.
A : Set of all Adult. The above Venn diagram represents the truth of the given statement, i.e. C ∩ A = Φ

#### QUESTION

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

#### SOLUTION

For x = 3, 3x + 2 = 3(3) + 2 = 9 + 2 = 11 > 9

∴ x = 3 satisfies the equation 3x + 2 > 9.

∴ The given statement is true.

∴ Its truth value is T.

#### QUESTION

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.

#### SOLUTION

For x = 5, x2 = 52 = 25 < 18

∴ x = 5 does not satisfies the equation x2 < 18.

∴ The given statement is false.

∴ Its truth value is F.

#### QUESTION

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.

#### SOLUTION

For x = 2, x + 3 = 2 + 3 = 5 < 11.

∴ x = 2 satisfies the equation x + 3 < 11.

∴ The given statement is true.

∴ Its truth value is T.

#### QUESTION

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.

#### SOLUTION

There is no x in A which satisfies x2 + 2 ≥ 5.

∴ The given statement is false.

∴ Its truth value is F.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

7 is prime number and Tajmahal is in Agra.

#### SOLUTION

Let p : 7 is prime number.
q : Tajmahal is in Agra.

The given statement in symbolic form is p ∧ q.

Its negation is ~(p ∧ q) ≡ ~p ∨ ~q.

∴ The negation of given statement is '7 is not prime number or Tajmahal is not in Agra.'

#### QUESTION

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

10 > 5 and 3 < 8

#### SOLUTION

Let p : 10 > 5.
q : 3 < 8.

The given statement in symbolic form is p ∧ q.

Its negation is ~(p ∧ q) ≡ ~p ∨ ~q.

∴ The negation of given statement is '10 ≤ 5 or 3 ≥ 8.'

#### QUESTION

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

I will have tea or coffee.

#### SOLUTION

Let p : I will have tea.
q : I will have coffee.

The given statement in symbolic form is p ∨ q.

Its negation is ~(p ∨ q) ≡ ~p ∧ ~q.

∴ The negation of given statement is ‘I will not have tea and coffee’.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

∀ n ∈ N, n + 3 > 9.

#### SOLUTION

∃ n ∈ N such that n + 3 ≤ 9.

#### QUESTION

Miscellaneous Exercise 1 | Q 4.22 | Page 34

Write the negation of the following statement.

∃ n ∈ A, such that x + 5 < 11.

#### SOLUTION

∀  x ∈ A, x + 5 ≤ 11