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

Asked by Aaryan | 2 years ago |  164

1 Answer

Solution :-

(i) Direct Method:

Let us assume that ‘q’ and ‘r’ be the statements given by

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

r: x is 0.

The given statement can be written as:

if q, then r.

Let q be true. Then, x is a real number such that x+ x = 0

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

x = 0

r is true

Thus, q is true

Therefore, q is true and r is true.

Hence, p is true.

(ii) Method of Contrapositive:

Let r be false. 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 and r is true

(iii) Method of Contradiction:

If possible, let p be false. Then,

P is not true

-p is true

-p (p => r) 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.

Answered by Sakshi | 2 years ago

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