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  • Write down the negation of following compound statements. (i)All rational numbers are real and complex.

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) \left | x \right | is equal to either x or – x.
(viii) 6 is divisible by 2 and 3.

Answers (1)

(i) It is a compound statement whose 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

Thus,(p \wedge q) =All rational numbers are real and complex.

& -(p \wedge q) = -p V -q =All rational nos. are neither complex nor real

(ii) It is a compound statement whose components are –

p: all real numbers are rational

-p: all real numbers are not rational

q: All real numbers are not irrational &

-q: All real numbers are not irrational

Thus, (p v q) = All real nos. are either rational or irrational.

&  -(p \wedge q) = -p V -q= All real numbers are neither rational or irrational.

(iii) It is a compound statement whose components are –

p: x = 2 is a root of Quadratic equation x^{2} - 5x + 6 = 0.

-p: x = 2 is not a root of Quadratic equation x^{2} - 5x + 6 = 0.

q: x = 3 is a root of Quadratic equation x^{2} - 5x + 6 = 0.

-q: x = 3 is not a root of Quadratic equation x^{2} - 5x + 6 = 0.

Thus, (p \wedge q) = x = 2 \; and\; x = 3 are roots of Quadratic equation x^{2} - 5x + 6 = 0.

& -(p \wedge q) = -p V -q

 Neither x = 2 nor x = 3 are roots of Quadratic equation x^{2} - 5x + 6 = 0.

(iv) It is a compound statement whose 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 sides

Thus, (p V q) = A triangle has either 3 or 4 sides  &-(p V q) = -p \wedge -q  = A triangle has neither 3 nor 4 sides

(v) It is a compound statement whose components are –

p: 35 is a prime no.

-p: 35 is not a prime no.

q: 35 is a composite no.

-q: 35 is not a composite no.

Thus, (p V q) = 35 is either a prime no. or a composite no. & -(p V q) = -p ? –q = 35 is neither a prime no. nor a composite no.

(vi) It is a compound statement whose components are –

p: All prime integers are even

-p: All prime integers are not even

q: All integers are odd

-q: All integers are not odd

Thus, (p V q) = All prime integers are either even or odd & -(p V q) = -p ? –q = All prime integers are neither even nor odd

(vii) It is a compound statement whose components are –

p: \left | x \right | is equal to x.

-p: \left | x \right | is not equal to x.

q: \left | x \right | is equal to-x.

-q: \left | x \right | is not equal to-x.

Thus, (p V q) = x is either equal to x or-x  & -(p V q) = -p ? – = \left | x \right | is neither equal to x nor-x.

(viii) ) It is a compound statement whose 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

Thus, (p ^ q) = 6 is divisible by 2 & 3

& -(p v q) = -p V –q  = 6 is neither divisible by 2 nor 3.

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