**Chapter 1 Mathematical Logic Ex 1.1**

**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.

** (iv) Polygon ABCDE is a pentagon.Solution:**
Polygon ABCDE is not a pentagon.

**(v) 7 + 3 > 5.Solution :**

7 + 3 > 5.