Question

Consider the statement :

P:if x a real number such that $x^3+4x=0$, then x=0

Prove that p is a true statement , using: (i) direct method (ii) method of contradiction (iii) method of contrapositive

Answer 1

(i) Direct method:

$let{x}^3+4x=0,$ where $x\in R,$then,

$x^3+4x=0\Rightarrow x(x^2+4)=0$

$\Rightarrow x=0[\because x^2+4\ne 0forx\in R]$

Hence, p is a true statement.

(ii) Method of contradiction

if possible,$let(x^3+4x=0andx\ne 0then,$

$x^2(x^2+4)=0andx\ne 0\Rightarrow x^2+4=0$

But, this is a contradiction, since$x^2+4\ne 0forx\in R$

Since, the contradiction arises by assuming that

$x^2+4=0andx\ne 0,sox=0$

Hence,$x^3+4x=0\Rightarrow 0$ is a true statement.

(iii) Method contrapositive:

We have to porve that $x^3+4x=0\Rightarrow x=0$

Let $P:x^3+4x=0andq:x=0$.