# 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]

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

#### SOLUTION

Come here

#### QUESTION

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

#### QUESTION

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

#### QUESTION

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

#### QUESTION

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

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### QUESTION

Choose the correct alternative :

The statement (∼ p ∧ q) ∨∼ q is

p ∨ q

p ∧ q

∼ (p ∨ q)

∼ (p ∧ q)

#### SOLUTION

#### QUESTION

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

#### QUESTION

Choose the correct alternative :

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

∼ p

p

q

∼ q

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### QUESTION

Choose the correct alternative :

Conditional p → q is equivalent to

p → ∼ q

∼ p ∨ q

∼ p → ∼ q

p ∨∼q

#### SOLUTION

#### QUESTION

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

#### QUESTION

Choose the correct alternative :

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

contingency

contradiction

tautology

None of them

#### SOLUTION

#### QUESTION

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

Fill in the blanks :

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

#### SOLUTION

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

#### QUESTION

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

Fill in the blanks :

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

#### SOLUTION

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

#### QUESTION

Fill in the blanks :

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

#### SOLUTION

Truth value of if x = 2, then x^{2} = − 4 is F.

#### QUESTION

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

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

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

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

State whether the following statement is True or False :

Truth value of 2 + 3 < 6 is F.

True

False

#### SOLUTION

#### QUESTION

State whether the following statement is True or False :

There are 24 months in year is a statement.

True

False

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

State whether the following statement is True or False :

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

True

False

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

State whether the following statement is True or False :

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

True

False

#### SOLUTION

#### QUESTION

State whether the following statement is True or False :

x^{2} = 25 is true statement.

True

False

#### SOLUTION

#### QUESTION

State whether the following statement is True or False :

Truth value of

True

False

#### SOLUTION

#### QUESTION

State whether the following statement is True or False :

p ∧ t = p.

True

False

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

Solve the following :

State which of the following sentences are statements in logic.

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

Is a statement

Is not a statement

#### SOLUTION

#### QUESTION

Solve the following :

State which of the following sentences are statements in logic.

Why are you sad?

Is a statement

Is not a statement

#### SOLUTION

#### QUESTION

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

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### QUESTION

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

#### QUESTION

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

#### QUESTION

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

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### QUESTION

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

#### SOLUTION

#### QUESTION

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

#### QUESTION

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

#### QUESTION

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

#### SOLUTION

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

Hence, its truth value is F.

#### QUESTION

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

#### QUESTION

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

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

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

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

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

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

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

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.

#### SOLUTION

Let p : x^{3} + y^{3} = (x + y)^{3}

q : xy = 0

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

#### QUESTION

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

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

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

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

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

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

If p : Proof is lengthy.

q : It is interesting.

Express the following statement in symbolic form.

Proof is lengthy and it is not interesting.

#### SOLUTION

p ∧ ∼ q

#### QUESTION

If p : Proof is lengthy.

q : It is interesting.

Express the following statement in symbolic form.

If proof is lengthy then it is interesting.

#### SOLUTION

p → q

#### QUESTION

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.

#### SOLUTION

∼(p ∧ q)

#### QUESTION

If p : Proof is lengthy.

q : It is interesting.

Express the following statement in symbolic form.

It is interesting iff the proof is lengthy.

#### SOLUTION

q ↔ p

#### QUESTION

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

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

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

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

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

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

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

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

Determine the truth value of the following statement.

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

#### SOLUTION

Let p : 9 > 1

q : x^{2} – 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

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.

#### SOLUTION

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

q : y^{2} = 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

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

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

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

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

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

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

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

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

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

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

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

∼ (p ∧ q) → ∼ (q ∧ p)

#### SOLUTION

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

≡ ~ T → ~ T

≡ F → F

≡ T

Hence, truth value is T.

#### QUESTION

∼ [(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

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

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

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

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

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

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

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

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

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

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

**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. - Contradiction:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Write the dual of the following.

(~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q)

#### SOLUTION

(~p ∨ q) ∧ (p ∨ ~q) ∧ (~p ∨ ~q)

#### QUESTION

Write the dual of the following.

(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

#### SOLUTION

(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

#### QUESTION

Write the dual of the following.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (q ∨ r)

#### SOLUTION

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (q ∧ r)

#### QUESTION

Write the dual of the following.

~(p ∨ q) ≡ ~p ∧ ~q

#### SOLUTION

~(p ∧ q) ≡ ~p ∨ ~q

#### QUESTION

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

#### QUESTION

#### QUESTION

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 = Î¦

#### QUESTION

#### QUESTION

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

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

∀ x ∈ A, x^{2} < 18.

#### SOLUTION

For x = 5, x^{2} = 5^{2} = 25 < 18

∴ x = 5 does not satisfies the equation x^{2} < 18.

∴ The given statement is false.

∴ Its truth value is F.

#### QUESTION

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

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

∀ x ∈ A, x^{2} + 2 ≥ 5.

#### SOLUTION

There is no x in A which satisfies x^{2} + 2 ≥ 5.

∴ The given statement is false.

∴ Its truth value is F.

#### QUESTION

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

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

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

Write the negation of the following statement.

∀ n ∈ N, n + 3 > 9.

#### SOLUTION

∃ n ∈ N such that n + 3 ≤ 9.

#### QUESTION

Write the negation of the following statement.

∃ n ∈ A, such that x + 5 < 11.

#### SOLUTION

∀ x ∈ A, x + 5 ≤ 11