Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf Chapter 1 Mathematical Logic Ex 1.4 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 1 Mathematical Logic Ex 1.4

Question 1.

Write the following statements in symbolic form:

(i) If the triangle is equilateral, then it is equiangular.

Solution:

Let p : Triangle is equilateral.

q : It is equiangular.

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

(ii) It is not true that ‘i’ is a real number.

Solution:

Let p : ‘i’ is a real number.

Then the symbolic form of the given statement is ~p.

(iii) Even though it is not cloudy, it is still raining.

Solution:

Let p : It is cloudy.

q : It is still raining.

Then the symbolic form of the given statement is ~p ∧ q.

(iv) Milk is white if and only if the sky is not blue.

Solution:

Let p : Milk is white.

q : Sky is blue.

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

(v) Stock prices are high if and only if stocks are rising.

Solution:

Let p : Stock prices are high.

q : stocks are rising.

Then the symbolic form of the given statement is p ↔ q

(vi) If Kutub-Minar is in Delhi, then Taj Mahal is in Agra.

Solution:

Let p : Kutub-Minar is in Delhi.

q : Taj Mahal is in Agra.

Then the symbolic form of the given statement is p → q

Question 2.

Find the truth value of each of the following statements:

(i) It is not true that 3 – 7i is a real number.

Solution:

Let p : 3 – 7i be a real number.

Then the symbolic form of the given statement is ~p.

The truth value of p is F.

∴ the truth value of ~p is T. ….[~F ≡ T]

(ii) If a joint venture is a temporary partnership, then a discount on purchase is credited to the supplier.

Solution:

Let p : Joint venture is a temporary partnership.

q : Discount on purchases is credited to the supplier.

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

The truth values of p and q are T and F respectively.

∴ the truth value of p → q is F. …..[T → F ≡ F]

(iii) Every accountant is free to apply his own accounting rules if and only if machinery is an asset.

Solution:

Let p : Every accountant is free to apply his own accounting rules.

q : Machinery is an asset.

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

The truth values of p and q are F and T respectively.

∴ the truth value of p ↔ q is F. ….[F ↔ T ≡ F]

(iv) Neither 27 is a prime number nor divisible by 4.

Solution:

Let p : 27 is a prime number.

q : 27 is divisible by 4.

Then the symbolic form of the given statement is ~p ∧ ~q.

The truth values of both p and q are F.

∴ the truth value of ~p ∧ ~q is T. …..[~F ∧ ~F ≡ T ∧ T ≡ T]

(v) 3 is a prime number and an odd number.

Solution:

Let p : 3 be a prime number.

q : 3 is an odd number.

Then the symbolic form of the given statement is p ∧ q

The truth values of both p and q are T.

∴ the truth value of p ∧ q is T. …..[T ∧ T ≡ T]

Question 3.

If p and q are true and r and s are false, find the true value of each of the following statements:

(i) p ∧ (q ∧ r)

Solution:

Truth values of p and q are T and truth values of r and s are F.

p ∧ (q ∧ r) ≡ T ∧ (T ∧ F)

≡ T ∧ F

≡ F

Hence, the truth value of the given statement is false.

(ii) (p → q) ∨ (r ∧ s)

Solution:

(p → q) ∨ (r ∧ s) ≡ (T → T) ∨ (F ∧ F)

≡ T ∨ F

≡ T

Hence, the truth value of the given statement is true.

(iii) ~[(~p ∨ s) ∧ (~q ∧ r)]

Solution:

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

≡ ~[(F ∨ F) ∧ (F ∧ F)]

≡ ~(F ∧ F)

≡ ~F

≡ T

Hence, the truth value of the given statement is true.

(iv) (p → q) ↔ ~(p ∨ q)

Solution:

(p → q) ↔ ~(p ∨ q) = (T → T) ↔ ~(T ∨ T)

≡ T ↔ ~ (T)

≡ T ↔ F

≡ F

Hence, the truth value of the given statement is false.

(v) [(p ∨ s) → r] ∨ [~(p → q) ∨ s]

Solution:

[(p ∨ s) → r] ∨ ~[~(p → q) ∨ s]

≡ [(T ∨ F) → F] ∨ ~[ ~(T → T) ∨ F]

≡ (T → F) ∨ ~(~T ∨ F)

≡ F ∨ ~ (F ∨ F)

≡ F ∨ ~F

≡ F ∨ T

≡ T

Hence, the truth value of the given statement is true.

(vi) ~[p ∨ (r ∧ s)] ∧ ~[(r ∧ ~s) ∧ q]

Solution:

~[p ∨ (r ∧ s)] ∧ ~[(r ∧ ~s) ∧ q]

≡ ~[T ∨ (F ∧ F)] ∧ ~[(F ∧ ~F) ∧ T]

≡ ~[T ∨ F] ∧ ~[(F ∧ T) ∧ T]

≡ ~T ∧ ~(F ∧ T)

≡ F ∧ ~F

≡ F ∧ T

≡ F

Hence, the truth value of the given statement is false.

Question 4.

Assuming that the following statements are true:

p : Sunday is a holiday.

q : Ram does not study on holiday.

Find the truth values of the following statements:

(i) Sunday is not holiday or Ram studies on holiday.

Solution:

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

∴ the truth value of the given statement is F.

(ii) If Sunday is not a holiday, then Ram studies on holiday.

Solution:

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

∴ the truth value of the given statement is T.

(iii) Sunday is a holiday and Ram studies on holiday.

Solution:

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

∴ the truth value of the given statement is F.

Question 5.

If p : He swims.

q : Water is warm.

Give the verbal statements for the following symbolic statements:

(i) p ↔ ~q

Solution:

p ↔ ~ q

He swims if and only if the water is not warm.

(ii) ~(p ∨ q)

Solution:

~(p ∨ q)

It is not true that he swims or water is warm.

(iii) q → p

Solution:

q → p

If water is warm, then he swims.

(iv) q ∧ ~p

Solution:

q ∧ ~p

The water is warm and he does not swim.