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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