Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 1 Mathematical Logic Ex 1.1 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 1.

State which of the following sentences are statements. Justify your answer. In case of statement, write down the truth value :

(i) 5 + 4 = 13.

Solution:

It is a statement which is false, hence its truth value is ‘F’.

(ii) x – 3 = 14.

Solution:

It is an open sentence, hence it is not a statement.

(iii) Close the door.

Solution:

It is an imperative sentence, hence it is not a statement.

(iv) Zero is a complex number.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(v) Please get me breakfast.

Solution:

It is an imperative sentence, hence it is not a statement.

(vi) Congruent triangles are also similar.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(vii) x^{2} = x.

Solution:

It is an open sentence, hence it is not a statement,

(viii) A quadratic equation cannot have more than two roots.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(ix) Do you like Mathematics ?

Solution:

It is an interrogative sentence, hence it is not a statement.

(x) The sun sets in the west.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(xi) All real numbers are whole numbers.

Solution:

It is a statement which is false, hence its truth value is ‘F’.

(xii) Can you speak in Marathi ?

Solution:

It is an interrogative sentence, hence it is not a statement.

(xiii) x^{2} – 6x – 7 = 0, when x = 7.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(xiv) The sum of cuberoots of unity is zero.

Solution:

It is a statement which is true, hence its truth value is ‘T’.

(xv) It rains heavily.

Solution :

It is an open sentence, hence it is not a statement.

Question 2.

Write the following compound statements symbolically:

(i) Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Solution:

Let p : Nagpur is in Maharashtra.

q : Chennai is in Tamil Nadu.

Then the symbolic form of the given statement is P∧q.

(ii) Triangle is equilateral or isosceles,

Solution:

Let p : Triangle is equilateral.

q : Triangle is isosceles.

Then the symbolic form of the given statement is P∨q.

(iii) The angle is right angle if and only if it is of measure 90°.

Solution:

Let p : The angle is right angle.

q : It is of measure 90°.

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

(iv) Angle is neither acute nor obtuse.

Solution:

Let p : Angle is acute.

q : Angle is obtuse.

Then the symbolic form of the given statement is

~p ∧ ~q.

(v) If ∆ ABC is right angled at B, then m∠A + m∠C = 90°.

Solution:

Let p : ∆ ABC is right angled at B.

q : m∠A + m∠C = 90°.

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

(vi) Hima Das wins gold medal if and only if she runs fast.

Solution:

Let p : Hima Das wins gold medal

q : She runs fast.

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

(vii) x is not irrational number but it is a square of an integer.

Solution:

Let p : x is not irrational number

q : It is a square of an integer

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

Note : If p : x is irrational number, then the symbolic form of the given statement is ~p ∧ q.

Question 3.

Write the truth values of the following :

(i) 4 is odd or 1 is prime.

Solution:

Let p : 4 is odd.

q : 1 is prime.

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 v q is F. … [F ∨ F = F]

(ii) 64 is a perfect square and 46 is a prime number.

Solution:

Let p : 64 is a perfect square.

q : 46 is a prime number.

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) 5 is a prime number and 7 divides 94.

Solution:

Let p : 5 is a prime number.

q : 7 divides 94.

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]

(iv) It is not true that 5 – 3i is a real number.

Solution:

Let p : 5 – 3i is a real number.

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

The truth values of p is F.

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

(v) If 3 × 5 = 8, then 3 + 5 = 15.

Solution:

Let p : 3 × 5 = 8.

q : 3 + 5 = 15.

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]

(vi) Milk is white if and only if sky is blue.

Solution:

Let p : Milk is white.

q : Sky is blue

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]

(vii) 24 is a composite number or 17 is a prime number.

Solution :

Let p : 24 is a composite number.

q : 17 is a prime 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 4.

If the statements p, q are true statements and r, s are false statements, then determine the truth values of the following:

(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 ≡ T

Hence the truth value of the given statement is true.

(ii) (p → q) ∨ (r → s)

Solution:

(p → q) ∨ (r → s) ≡ (T → T) ∨ (F → F)

≡ T ∨ T ≡ T

Hence the truth value of the given statement is true.

(iii) (q ∧ r) ∨ (~p ∧ s)

Solution:

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

≡ F ∨ (F ∧ F)

≡ F ∨ F ≡ F

Hence the truth value of the given statement is false.

(iv) (p → q) ∧ (~ r)

Solution:

(p → q) ∧ (~ r) ≡ (T → T) ∧ (~ F)

≡ T ∧ T ≡ T

Hence the truth value of the given statement is true.

(v) (~r ↔ p) → (~q)

Solution:

(~r ↔ p) → (~q) ≡ (~F ↔ T) → (~T)

≡ (T ↔ T) → F

≡ T → F ≡ F

Hence the truth value of the given statement is false.

(vi) [~p ∧ (~q ∧ r) ∨ (q ∧ r) ∨ (p ∧ r)]

Solution:

[~p ∧ (~q ∧ r)∨(q ∧ r)∨(p ∧ r)]

≡ [~T ∧ (~T ∧ F)] ∨ [(T ∧ F) V (T ∧ F)]

≡ [F ∧ (F ∧ F)] ∨ [F V F]

≡ (F ∧ F) ∨ F

≡ F ∨ F ≡ F

Hence the truth value of the given statement is false.

(vii) [(~ p ∧ q) ∧ (~ r)] ∨ [(q → p) → (~ s ∨ r)]

Solution:

[(~ p ∧ q) ∧ (~ r)] ∨ [(q → p) → (~ s ∨ r)]

≡ [(~T ∧ T) ∧ (~F)] ∨ [(T → T) → (~F ∨ F)]

≡ [(F ∧ T) ∧ T] ∨ [T → (T ∨ F)]

≡ (F ∧ T) ∨ (T → T)

≡ F ∨ T ≡ T

Hence the truth value of the given statement is true.

(viii) ~ [(~p ∧ r) ∨ (s → ~q)] ↔ (p ∧ r)

Solution :

~ [(~p ∧ r) ∨ (s → ~q)] ↔ (p ∧ r)

≡ ~ [(~T ∧ F) ∨ (F → ~T)] ↔ (T ∧ F)

≡ ~ [(F ∧ F) ∨ (F → F)] ↔ F

≡ ~ (F ∨ T) ↔ F

≡ ~T ↔ F

≡ F ↔ F ≡ T

Hence the truth value of the given statement is true.

Question 5.

Write the negations of the following :

(i) Tirupati is in Andhra Pradesh.

Solution:

The negations of the given statements are :

Tirupati is not in Andhra Pradesh.

(ii) 3 is not a root of the equation x^{2} + 3x – 18 = 0.

Solution:

3 is a root of the equation x^{2} + 3x – 18 = 0.

(iii) \(\sqrt {2}\) is a rational number.

Solution:

\(\sqrt {2}\) is not a rational number.

(iv) Polygon ABCDE is a pentagon.

Solution:

Polygon ABCDE is not a pentagon.

(v) 7 + 3 > 5.

Solution :

7 + 3 > 5.