Write down the negative 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 traingle has either 3-sides or 4- sides.
(v) 35 is a prime number or a composite cumber .
(vi) All prime integers are either even or odd.
(viii) |x| is equl to either x or x .
(viii) 6 is divisible by 2 and 3.

Answer

N/a

Explanation

(i) Let p : All rational numbers are real.
q: All rational numbers are complex.
~q :All raticnal number are not real.
~p: All ration al numbers are not complex
`~(p^^q)` : All rational numbers are not real or not complex.`" "[because ~(p^^q) =~ pv ~q]`
(ii) Let p: All real numbers are rationals.
q: All real numbers are irrational.
Then, the negation of the above statements is given by .
`~(pveeq)`: All real numbers are not rational and all real numbers are not irrational`" "[because ~( p veeq) = ~p ^^~q]`.
(iii) Let p: 2 is root of quadratic equation `x^(2)-5x + 6= 0`
q : x = 3 is root of quadratic equation `x^(2)-5x + 6= 0.`
Then, the negation of conjunction of above staterment is gven by .
` ~( p veeq) : x =2` is not a root of quadratic equation `x^(2)-5x+ 6= 0 or x =3` is not a root of the quadriatic equation `x^(2) - 5x + 6=0`
(iv) Let p: A triangle has 3-sides.
q: A triangle has 4-sides.
Then, negation of disjunction of the above statement is given by.
`~ ( p veeq)` : A triangle has neither 3-sides nor 4 -sides.
(v) Let p: 35 is a prime number
q:35 is a cormposite number.
Then, negaton of cisjurcton of the above statement is given by
`~(p veeq)` : 35 is not a prime number and it is not a composite number.
(vi) Let p: All prime integers are even.
q: All prime integre ar odd.
Then negation f disjuncticn of the above statement is givon by
`~ (q veeq)` : 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 negation of disjunction of the above statement is given by
`~ (q veeq)` : |X| is not equal to x and it is not equal to -x .
(viii) Let p: 6 is divisible by 2.
q: 6 is divisble by 3.
Then, negaton of conjunction of aoove statement is given by
`~ (q veeq)` : 6 is not divisible by 2 or it is not divisible by 3