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

Write the negation of the following statement.

All the stars are shining if it is night.

#### SOLUTION

Let q : All stars are shining.

p : It is night.

The given statement in symbolic form is p → q. It’s negation is ~ (p → q) ≡ p ∧ ~ q

∴ The negation of a given statement is ‘It is night and some stars are not shining’.

Write the negation of the following statement.

∀ n ∈ N, n + 1 > 0

#### SOLUTION

∃ n ∈ N such that n + 1 ≤ 0.

Write the negation of the following statement.

∃ n ∈ N, (n^{2} + 2) is odd number.

#### SOLUTION

∀ n ∈ N, (n^{2} + 2) is not odd number.

Write the negation of the following statement.

Some continuous functions are differentiable.

#### SOLUTION

All continuous functions are not differentiable.

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

(p → r) ∧ q

#### SOLUTION

~ [(p → r) ∧ q] ≡ ~(p → r) ∨ ~q ....[Negation of conjunction]

≡ (p ∧ ~ r) ∨ ~q .....[Negation of implication]

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

~(p ∨ q) → r

#### SOLUTION

~[~(p ∨ q) → r] ≡ ~(p ∨ q) ∧ ~r ....[Negation of implication]

≡ (~p ∧ ~q) ∧ ~r .....[Negation of disjunction]

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

(~p ∧ q) ∧ (~q ∨ ~r)

#### SOLUTION

~[(~p ∧ q) ∧ (~q ∨ ~r)]

≡ ~(~ p ∧ q) ∨ ~ (~ q ∨ ~r) ...[Negation of conjunction]

≡ [~(~ p) ∨ ~ q] ∨ [~(~q) ∧ ~(~r)] ...[Negation of conjunction and disjunction]

≡ (p ∨ ~q) ∨ (q ∨ r) .....[Negation on negation]

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

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

#### SOLUTION

Let p : It snows.

q : They do not drive the car.

∴ The given statement is p → q.

Its converse is q → p.

If they do not drive the car then it snows.

Its inverse is ~p → ~q.

If it does not snow then they drive the car.

Its contrapositive is ~q → ~p.

If they drive the car then it does not snow.

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

If he studies, then he will go to college.

#### SOLUTION

Let p : He studies.

q : He will go to college.

∴ The given statement is p → q.

Its converse is q → p.

If he will go to college then he studies.

Its inverse is ~p → ~q.

If he does not study then he will not go to college.

Its contrapositive is ~q → ~p.

If he will not go to college then he does not study.

With proper justification, state the negation of the following.

(p → q) ∨ (p → r)

#### SOLUTION

~[(p → q) ∨ (p → r)]

≡ ~(p → q) ∧ ~(p → r) ...[Negation of disjunction]

≡ (p ∧ ~ q) ∧ (p ∧ ~r) ....[Negation of implication]

With proper justification, state the negation of the following.

(p ↔ q) ∨ (~q → ~r)

#### SOLUTION

~[(p ↔ q) ∨ (~q → ~r)]

≡ ~(p ↔ q) ∧ (~q → ~r) ....[Negation of disjunction]

≡ [(p ∧ ~q) ∨ (q ∧ ~p)] ∧ ~(~q → ~r) ....[Negation of double implication]

≡ [(p ∧ ~q) ∨ (q ∧ ~p)] ∧ [~ q ∧ ~(~r)] ....[Negation of implication]

≡ [(p ∧ ~q) ∨ (q ∧ ~p)] ∧ (~ q ∧ r) ....[Negation of negation]

With proper justification, state the negation of the following.

(p → q) ∧ r

#### SOLUTION

~[(p → q) ∧ r]

≡ ~ (p → q) ∨ ~ r ....[Negation of conjunction]

≡ (p ∧ ~q) ∨ ~ r ....[Negation of implication]