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 x² -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.

Answer 1

(i) Let p: All rational numbers are real.

q: All rational numbers are complex.

~ p: All rational numbers are not real.

~ q ; All rational numbers are not complex.

Then, the negation of the given compound statement is:

~ (p ∧ q): All rational numbers are not real or not complex.

[~(p ∧ q) = ~p v ~q]

(ii) Let p: All real numbers are rationals. q: All real numbers are irrationals. Then, the negation of the given compound statement is:

~ (p v q): All real numbers are not rational and all real numbers are not irrational. [~(p v q) = ~p ∧ ~ q]

(iii) Let p ; x = 2 is root of quadratic equation x² – 5x + 6 = 0. q: x = 3 is root of quadratic equation x² – 5x + 6 = 0.

Then, the negation of the given compound statement is:

~ (p ∧ q) : x = 2 is not a root of quadratic equation x² – 5x + 6 = 0 or x = 3 is not a root of the quadratic equation x² – 5x + 6 = 0.

(iv) Let p: A triangle has 3-sides. q: A triangle has 4-sides.

Then, the negation of the given compound statement is:

~ (p v q): A triangle has neither 3-sides nor 4-sides.

(v) Let p: 35 is a prime number. q: 35 is a composite number.

Then, the negation of the given compound statement is:

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

(vi) Let p: All prime integers are even. q: All prime integers are odd.

Then, the negation of the given compound statement is given by

~(p v q): All prime integers are not even and all prime integers are not odd.

(vii) Let p:|x| is equal to x. q: |x| is equal to —x.

Then, the negation of the given compound statement is:

~ (p v q): |x| is not equal to JC and it is not equal to —x.

(viii) Let p: 6 is divisible by 2. 

q: 6 is divisible by 3.

Then, the negation of the given compound statement is:

~ (p∧q): 6 is not divisible by 2 or it is not divisible by 3