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

Q: 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)

Solution

(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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