**Question:**

**Write down the negation of following compound statements**

**(i) All rational numbers are real and complex.**

**(ii) All real numbers are rationals or irrationals.**

**(iii) x = 2 and x = 3 are roots of the Quadratic equation x 2 – 5x + 6 = 0.**

**(iv) A triangle has either 3-sides or 4-sides.**

**(v) 35 is a prime number or a composite number.**

**(vi) All prime integers are either even or odd.**

**(vii) |x| is equal to either x or – x.**

**(viii) 6 is divisible by 2 and 3.**

**Solution:**

**(i) All rational numbers are real and complex.**

The given statement is compound statement then components are,

P: All rational numbers are real.

~p: All rational numbers are not real.

q: All rational numbers are complex.

~q: All rational numbers are not complex.

(p ᴧ q)= All rational numbers are real and complex.

~ (p ᴧ q)=~p v ~q= All rational numbers are neither real nor complex.

**(ii) All real numbers are rationals or irrationals.**

The given statement is compound statement then components are,

P: All real numbers are rational.

~p: All real numbers are not rational.

q: All real numbers are irrational.

~q: All real numbers are not irrational.

(p ᴧ q)= All real numbers are rationals or irrationals.

~(p ᴧ q)=~p v ~q= All real numbers are neither rationals nor irrationals.

**(iii) x = 2 and x = 3 are roots of the Quadratic equation x 2 – 5x + 6 = 0.**

The given sentence is a compound statement in which components are

p: x = 2 is a root of Quadratic equation x2 – 5x + 6 = 0.

~p: x = 2 is not a root of Quadratic equation x2 – 5x + 6 = 0.

q: x = 3 is a root of Quadratic equation x2 – 5x + 6 = 0.

~q: x = 3 is not a root of Quadratic equation x2 – 5x + 6 = 0.

(p ᴧ q)= x = 2 and x = 3 are roots of the Quadratic equation x2 – 5x + 6 = 0.

~ (p ᴧ q)=~p v ~q= Neither x = 2 and nor x = 3 are roots of x2 – 5x + 6 = 0

**(iv) A triangle has either 3-sides or 4-sides.**

The given statement is compound statement then components are,

P: A triangle has 3 sides

~p: A triangle does not have 3 sides.

q: A triangle has 4 sides.

~q: A triangle does not have 4 side.

(p v q)= A triangle has either 3-sides or 4-sides.

~(p v q)=~p ᴧ ~q= A triangle has neither 3 sides nor 4 sides.

**(v) 35 is a prime number or a composite number.**

The given statement is compound statement then components are,

P: 35 is a prime number

~p: 35 is not a prime number.

q: 35 is a composite number

~q: 35 is not a composite number.

(p v q)= 35 is a prime number or a composite number.

~ (p v q) = ~p ᴧ ~q = 35 is not a prime number and it is not a composite number.

**(vi) All prime integers are either even or odd.**

The given statement is compound statement then components are,

P: All prime integers are even

~p: All prime integers are not even.

q: All prime integers are odd

~q: All prime integers are not odd.

(p v q)= All prime integers are either even or odd.

~ (p v q)= ~p ᴧ ~q= All prime integers are not even and not odd.

**(vii) |x| is equal to either x or – x.**

The given statement is compound statement then components are,

P: |x| is equal to x.

~p: |x| is not equal to x.

q: |x| is equal to –x.

~q: |x| is not equal to -x.

(p v q)= |x| is equal to either x or – x.

~ (p v q) = ~p ᴧ ~q= |x| is not equal to x and |x| is not equal to – x.

**(viii) 6 is divisible by 2 and 3.**

The given statement is compound statement then components are,

P: 6 is divisible by 2

~p: 6 is not divisible by 2

q: 6 is divisible by 3

~q: 6 is not divisible by 3.

(p ᴧ q)= 6 is divisible by 2 and 3.

~ (p ᴧ q) = ~p v ~q= 6 is neither divisible by 2 nor 3