Question

Show that the statement

p : "If x is a real number such that $x^3+4x=0$ then x is 0" is true by (i) direct method (ii) Method of contrapositive.

Answer 1

The given compound statement is of the form "if p then q".

$p:xϵR$ such that $x^3+4x=0$

$q:x=0$

(i) Direct method :

We assume that p is true then

$xϵR$ such that $x^3+4x=0$

$\Rightarrow xϵR$ such $x(x^2+4)=0$

$\Rightarrow xϵR$ such that x = 0 or $x^2+4=0$

$\Rightarrow x=0$

$\Rightarrow q$ is true

So when p is true, q is true.

Thus the given compound statement is true.

(ii) Method of contradiction :

We assume that p is true and q is false.

then

$xϵR$ such that $x^3+4x=0$

$\Rightarrow xϵR$ such $x(x^2+4)=0$

$\Rightarrow xϵR$ such that $x=0$ or $x^2+4=0$

Since we assumed that $x\ne 0$, $x^2+4$ must be equal to 0. But $x^2+4$ is positive for any real x. This is a contradiction. So our assumption that $x\ne 0$ must be false. Thus the given compound statement is true.

(iii) Method of contrapositive :

We assume that q is false, then

$x\ne 0$

$\Rightarrow xϵR$ such that $x^3+4x\ne 0$

$\Rightarrow p$ is false

So when q is false, p is false.

Thus the given compound statement is true.