Show that the statement
p : “If x is a real number such that x3 + x = 0, then x is 0” is true by
(i) Direct method
(ii) method of contrapositive
(iii) method of contradiction
(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, x∈R
x(x2+1)≠0, x∈R
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