### Mathematical Logic - Ex 1.3 | Maharashtra Board 12th Maths Solutions Chapter 1

### Mathematical Logic - Ex 1.3 | Maharashtra Board 12th Maths Solutions Chapter 1

**Question 1.**

**(i) ÆŽ x ∈ A such that x – 8 = 1**

**Solution:**

Clearly x = 9 ∈ A satisfies x – 8 = 1.

So the given statement is true, hence its truth value is T.

**(ii) â±¯ x ∈ A, x2 + x is an even number**

**Solution:**

For each x ∈ A, x2 + x is an even number.

So the given statement is true, hence its truth value is T.

**(iii) ÆŽ x ∈ A such that x2 < 0**

**Solution:**

There is no x ∈ A which satisfies x2 < 0.

So the given statement is false, hence its truth value is F.

**(iv) â±¯ x ∈ A, x is an even number**

**Solution:**

x = 3 ∈ A, x = 5 ∈ A, x = 7 ∈ A, x = 9 ∈ A, x = 11 ∈ A do not satisfy x is an even number.

So the given statement is false, hence its truth value is F.

**(v) ÆŽ x ∈ A such that 3x + 8 > 40**

**Solution:**

Clearly x = 11 ∈ A and x = 12 ∈ A satisfies 3x + 8 > 40.

So the given statement is true, hence its truth value is T.

**(vi) â±¯ x ∈ A, 2x + 9 > 14**

**Solution:**

For each x ∈ A, 2x + 9 > 14. So the given statement is true, hence its truth value is T.

### Question 2.Write the duals of each of the following.

(i) p ∨ (q ∧ r)

**Solution:**

**The duals of the given statement patterns are :**

**p ∧ (q ∨ r)**

(ii) p ∧ (q ∧ r)

**Solution:**

**p ∨ (q ∨ r)**

(iii) (p ∨ q) ∧ (r ∨ s)

**Solution:**

**(p ∧ q) ∨ (r ∧ s)**

(iv) p ∧ ~q

**Solution:**

**p ∨ ~q**

(v) (~p ∨ q) ∧ (~r ∧ s)

**Solution:**

**(~p ∧ q) ∨ (~r ∨ s)**

(vi) ~p ∧ (~q ∧ (p ∨ q) ∧ ~r)

**Solution:**

**~p ∨ (~q ∨ (p ∧ q) ∨ ~r)**

(vii) [~(p ∨ q)] ∧ [p ∨ ~(q ∧ ~s)]

**Solution:**

**[ ~(p ∧ q)] ∨ [p ∧ ~(q ∨ ~s)]**

(viii) c ∨ {p ∧ (q ∨ r)}

**Solution:**

**t ∧ {p ∧ (q Ar)}**

(ix) ~p ∨ (q ∧ r) ∧ t

**Solution:**

**~p ∧ (q ∨ r) ∨ c**

(x) (p ∨ q) ∨ c

**Solution:**

**(p ∧ q) ∧ t**

### Question 3.Write the negations of the following.

**(i) x + 8 > 11 or y – 3 = 6**

**Solution:**

Let p : x + 8 > 11, q : y — 3 = 6.

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

Since ~(p ∨ q) ≡ ~p ∧ ~q, the negation of given statement is :

‘x + 8 > 11 and y – 3 ≠ 6’ OR

‘x + 8 ≮ 11 and y – 3 ≠ 6’

**(ii) 11 < 15 and 25 > 20**

**Solution:**

Let p: 11 < 15, q : 25 > 20.

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

Since ~(p ∧ q) ≡ ~p ∨ ~q, the negation of given statement is :

’11 ≮ 15 or 25 > 20.’ OR

’11 ≯ 15 or 25 ≮ 20.’

**(iii) Qudrilateral is a square if and only if it is a rhombus.**

**Solution:**

Let p : Quadrilateral is a square.

q : It is a rhombus.

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

Since ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p), the negation of given statement is :

‘ Quadrilateral is a square but it is not a rhombus or quadrilateral is a rhombus but it is not a square.’

**(iv) It is cold and raining.**

**Solution:**

Let p : It is cold.

q : It is raining.

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 not cold or not raining.’

**(v) If it is raining then we will go and play football.**

**Solution:**

Let p : It is raining.

q : We will go.

r : We play football.

Then the symbolic form of the given statement is p → (q ∧ r).

Since ~[p → (q ∧ r)] ≡ p ∧ ~(q ∧ r) ≡ p ∧ (q ∨ ~r), the negation of the given statement is :

‘It is raining and we will not go or not play football.’

**(vi) 2–√ is a rational number.**

**Solution:**

Let p : 2–√ is a rational number.

The negation of the given statement is

‘ ~p : 2–√ is not a rational number.’

**(vii) All natural numbers are whole numers.**

**Solution:**

The negation of the given statement is :

‘Some natural numbers are not whole numbers.’

**(viii) â±¯ n ∈ N, n2 + n + 2 is divisible by 4.**

**Solution:**

The negation of the given statement is :

‘ÆŽ n ∈ N, such that n2 + n + 2 is not divisible by 4.’

**(ix) ÆŽ x ∈ N such that x – 17 < 20**

**Solution:**

The negation of the given statement is :

‘â±¯ x ∈ N, x – 17 ≯ 20.’

### Question 4.Write converse, inverse and contrapositive of the following statements.

**(i) If x < y then x2 < y2 (x, y ∈ R)**

**Solution:**

Let p : x < y, q : x2 < y2.

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

Converse : q → p is the converse of p → q.

i.e. If x2 < y2, then x < y.

Inverse : ~p → ~q is the inverse of p → q.

i.e. If x ≯ y, then x2 ≯ y2. OR

If x ≮ y, then x2 ≮ y2.

Contrapositive : ~q → p is the contrapositive of

p → q i.e. If x2 ≯ y2, then x ≯ y. OR

If x2 ≮ y2, then x ≮ y.

**(ii) A family becomes literate if the woman in it is literate.**

**Solution:**

**Let p :**The woman in the family is literate.

**q :**A family become literate.

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

**Converse**: q → p is the converse of p → q.

i.e. If a family become literate, then the woman in it is literate.

**Inverse**: ~p → ~q is the inverse of p → q.

i.e. If the woman in the family is not literate, then the family does not become literate.

**Contrapositive :**~q → ~p is the contrapositive of p → q. i e. If a family does not become literate, then the woman in it is not literate.

**(iii) If surface area decreases then pressure increases.**

**Solution:**

**Let p :**The surface area decreases.

**q :**The pressure increases.

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

**Converse**: q → p is the converse of p→ q.

**i.e. I**f the pressure increases, then the surface area decreases.

**Inverse**: ~p → ~q is the inverse of p → q.

**i.e.**If the surface area does not decrease, then the pressure does not increase.

**Contrapositive**: ~q → ~p is the contrapositive of p → q.

**i.e.**If the pressure does not increase, then the surface area does not decrease.

**(iv) If voltage increases then current decreases.**

**Solution:**

**Let p :**Voltage increases.

**q**: Current decreases.

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

**Converse**: q →p is the converse of p → q.

i

**.e.**If current decreases, then voltage increases.**Inverse**: ~p → ~q is the inverse of p → q.

**i.e.**If voltage does not increase, then current does not decrease.

**Contrapositive**: ~q → ~p, is the contrapositive of p → q.

**i.e.**If current does not decrease, then voltage doesnot increase.

### Mathematical Logic - Ex 1.3 | Maharashtra Board 12th Maths Solutions Chapter 1

**Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 1 Mathematical Logic Ex 1.3 Questions and Answers.**- maharashtra board class 12 maths solutions part 1
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