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

EXERCISE 1.8 [PAGE 21]

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

EXERCISE 1.8Q 1.1   PAGE 21

Exercise 1.8 | Q 1.1 | Page 21

Write the negation of the following statement.

All the stars are shining if it is night.

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’.

EXERCISE 1.8Q 1.2   PAGE 21

Exercise 1.8 | Q 1.2 | Page 21

Write the negation of the following statement.

∀ n ∈ N, n + 1 > 0

SOLUTION

∃ n ∈ N such that n + 1 ≤ 0.

EXERCISE 1.8Q 1.3   PAGE 21

Exercise 1.8 | Q 1.3 | Page 21

Write the negation of the following statement.

∃ n ∈ N, (n² + 2) is odd number.

SOLUTION

∀ n ∈ N, (n² + 2) is not odd number.

EXERCISE 1.8Q 1.4   PAGE 21

Exercise 1.8 | Q 1.4 | Page 21

Write the negation of the following statement.

Some continuous functions are differentiable.

SOLUTION

All continuous functions are not differentiable.

EXERCISE 1.8Q 2.1   PAGE 21

Exercise 1.8 | Q 2.1 | Page 21

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]

EXERCISE 1.8Q 2.2   PAGE 21

Exercise 1.8 | Q 2.2 | Page 21

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]

EXERCISE 1.8Q 2.3   PAGE 21

Exercise 1.8 | Q 2.3 | Page 21

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]

EXERCISE 1.8Q 3.1   PAGE 21

Exercise 1.8 | Q 3.1 | Page 21

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

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

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.

EXERCISE 1.8Q 3.2   PAGE 21

Exercise 1.8 | Q 3.2 | Page 21

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

If he studies, then he will go to college.

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.

EXERCISE 1.8Q 4.1   PAGE 21

Exercise 1.8 | Q 4.1 | Page 21

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]

EXERCISE 1.8Q 4.2   PAGE 21

Exercise 1.8 | Q 4.2 | Page 21

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]

EXERCISE 1.8Q 4.3   PAGE 21

Exercise 1.8 | Q 4.3 | Page 21

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]