Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf

Chapter 1 Mathematical Logic Ex 1.8 Questions and Answers.

Question 1.
Write the negation of each of the following statements:
(i) All the stars are shining if it is night.
Solution:
The given statement can be written as:
If it is night, then all the stars are shining.
Let p : It is night.
q : All the stars are shining.
Then the symbolic form of the given statement is p → q
Since, ~(p → q) ≡ p ∧ ~q,
the negation of the given statement is ‘It is night and all the stars are not shining.’

(ii) ∀ n ∈ N, n + 1 > 0.
Solution:
The negation of the given statement is
‘∃ n ∈ N, such that n + 1 ≤ 0.’

(iii) ∃ n ∈ N, such that (n² + 2) is odd number.
Solution:
The negation of the given statement is
‘∀ n ∈ N, n² + 2 is not an odd number.’

(iv) Some continuous functions are differentiable.
Solution:
The negation of a given statement is ‘All continuous functions are not differentiable.’



Question 2.
Using the rules of negation, write the negations of the following:
(i) (p → r) ∧ q
Solution:
The negation of (p → r) ∧ q is
~[(p → r) ∧ q] ≡ ~(p → r) ∨ (~q) …..[Negation of conjunction]
≡ (p ∧ ~r) ∨ (~q) ……[Negation of implication]

(ii) ~(p ∨ q) → r
Solution:
The negation of ~(p ∨ q) → r is
~[~(p ∨ q) → r] ≡ ~(p ∨ q) ∧ (~r) …..[Negation of implication]
≡ (~p ∧ ~q) ∧ (~r) ……[Negation of disjunction]

(iii) (~p ∧ q) ∧ (~q ∨ ~r)
Solution:
The negation of (~p ∧ q) ∧ (~q ∨ ~r) is
~[(~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 of negation]

Question 3.
Write the converse, inverse, and contrapositive of the following statements:
(i) If it snows, then they do not drive the car.
Solution:
Let p : It snows.
q : They do not drive the car.
Then the symbolic form of the given statement is p → q.
Converse: q → p is the converse of p → q.
i.e. If they do not drive the car, then it snows.
Inverse: ~p → ~q is the inverse of p → q.
i.e. If it does not snow, then they drive the car.
Contrapositive: ~q → ~p is the contrapositive of p → q.
i.e. If they drive the car, then it does not snow.

(ii) If he studies, then he will go to college.
Solution:
Let p : He studies.
q : He will go to college.
Then two symbolic form of the given statement is p → q.
Converse: q → p is the converse of p → q.
i.e. If he will go to college, then he studies.
Inverse: ~p → ~q is the inverse of p → q.
i.e. If he does not study, then he will not go to college.
Contrapositive: ~q → ~p is the contrapositive of p → q.
i.e. If he will not go to college, then he does not study.



Question 4.
With proper justification, state the negation of each of the following:
(i) (p → q) ∨ (p → r)
Solution:
The negation of (p → q) ∨ (p → r) is
~[(p → q) ∨ (p → r)] ≡ ~(p → q) ∧ ~(p → r) …..[Negation of disjunction]
≡ (p ∧ ~q) ∧ (p ∧ ~r) …[Negation of implication]

(ii) (p ↔ q) ∨ (~q → ~r)
Solution:
The negation of (p ↔ q) ∨ (~q → ~r) is
~[(p ↔ q) ∨ (~q → ~r)] ≡ ~(p ↔ q) ∧ ~(~q → ~r) …..[Negation of disjunction]
≡ [(p ∧ ~q) ∨ (q ∧ ~p)] ∧ [~q ∧ ~(~r)] ……[Negation of biconditional and implication]
≡ [(p ∧ ~q) ∨ (q ∧ ~p)] ∧ (~q ∧ r) ……[Negation of negation]

(iii) (p → q) ∧ r
Solution:
The negation of (p → q) ∧ r is
~[(p → q) ∧ r] ≡ ~(p → q) ∨ (~r) …..[Negation of conjunction]
≡ (p ∧ ~q) ∨ (~r) …..[Negation of implication]