Answer :

(i) Direct Method:

Let us Assume that q and r be the statements given by

q: x is a real number such that x3+x=0.

r: x is 0.

since, the given statement can be written as :

if q, then r.

Let q be true . then,

x is a real number suc that x3+x = 0

x is a real number such that x(x2+1) = 0

x = 0

r is true

Thus, q is true

Therefore, q is true r is true

Hence, p is true.

(ii). Method of contrapositive

Let r be not true. then,

R is not true

x ≠ 0, xR

x(x2+1)≠0, xR

q is not true

Thus, -r = -q

Hence, p : q r is true

(iii) Method of Contradiction

If possible, let p be not true. Then,

P is not true

-p is true

-p(pr) is true

q and –r is true

x is a real number such that x3+x = 0and x≠ 0

x =0 and x≠0

This is a contradiction

Hence, p is true

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