Negate the following: (1) If an integer is greater than 3 and less th

1.	Negate the following: (1) If an integer is greater than 3 and less than 5 then it is a multiple of 5. (ii) If ‘x’ is divisible by ‘y’ then it is divisible by ‘a’ and ‘b. (iii) Weather is fine and my friends are not coming or we do not go to a movie. (iv) If a triangle is equilateral then it’s sides are equal and angles are equal. (v) 14 is a divisor of 48 and 28 is not divisible by 82.
Answer» (i) Let p: An integer is greater than 3 q: An integer is less than 5 r: An integer is multiple of 5 Given[(p ∧ q) → r] ∴ ~ ((p ∧ q) → r] = (p ∧ q) ∧~r An integer greater than 3 and less than 5 and not divisible by b (ii) Let P:x is divisible by y q: x is divisible by a r: x is divisible by b Given [p → (q ∧ r)] ∴ ~P (q ∧ r)] = p ∧~(q ∧ ~r) = p ∧( ~ ( q ∨ ~r) x is divisible by y and x is not divisible by a or not a multiple of 5 (iii) Let p: Weather is fine q: Friends are coming r: we go to a movie Given[p∧~q ∨~r)] ~(p ∧ (~q ∨ ~r)) ≡ ~p ∨ ~(~ q ∨ ~r) : ≡ ~p ∨ (q ∧ r) (iv) Let p = a triangle is equilateral q: Sides are equal r: angles are equal Given p → (q ∧ r) ∴ ~(p → (~q ∨ ~r)) ≡ ~p ∧ ~q ∨ ~r) A triangle is equilateral and sides are not. equal or angles are not equal. Let p: 14 is a divisor of 48 q: 28 is divisible by 82 Given p ∧ ~q ∴ ~(p ∧ ~q) ≡ ~p ∨ q 14 is not a divisor of 48 or 28 is divisible by 82

Negate the following: (1) If an integer is greater than 3 and less than 5 then it is a multiple of 5. (ii) If ‘x’ is divisible by ‘y’ then it is divisible by ‘a’ and ‘b. (iii) Weather is fine and my friends are not coming or we do not go to a movie. (iv) If a triangle is equilateral then it’s sides are equal and angles are equal. (v) 14 is a divisor of 48 and 28 is not divisible by 82.

Answer»

(i) Let p: An integer is greater than 3

q: An integer is less than 5

r: An integer is multiple of 5

Given[(p ∧ q) → r]

∴ ~ ((p ∧ q) → r] = (p ∧ q) ∧~r

An integer greater than 3 and less than 5 and not divisible by b

(ii) Let P:x is divisible by y

q: x is divisible by a

r: x is divisible by b

Given [p → (q ∧ r)]

∴ ~P (q ∧ r)] = p ∧~(q ∧ ~r)

= p ∧( ~ ( q ∨ ~r)

x is divisible by y and x is not divisible by a or not a multiple of 5

(iii) Let p: Weather is fine

q: Friends are coming

r: we go to a movie

Given[p∧~q ∨~r)] ~(p ∧ (~q ∨ ~r)) ≡ ~p ∨ ~(~ q ∨ ~r) : ≡ ~p ∨ (q ∧ r)

(iv) Let p = a triangle is equilateral

q: Sides are equal

r: angles are equal

Given p → (q ∧ r)

∴ ~(p → (~q ∨ ~r)) ≡ ~p ∧ ~q ∨ ~r)

A triangle is equilateral and sides are not. equal or angles are not equal.

Let p: 14 is a divisor of 48

q: 28 is divisible by 82

Given p ∧ ~q

∴ ~(p ∧ ~q) ≡ ~p ∨ q

14 is not a divisor of 48 or 28 is divisible by 82