1.(iii)    Show that the statement             p: “If $x$ is a real number such that $x^3 + 4x = 0$, then $x$ is 0” is true by             (iii) method of contrapositive

If $x$ is a real number such that $x^3 + 4x = 0$, then $x$ is 0 : (if p then q)

p: x is a real number such that $\dpi{100} x^3 + 4x = 0$.

q: x is 0.

In order to prove the statement “if p then q”

Contrapositive Method:  By assuming that q is false, prove that p must be false.

So,

q is false: $\dpi{100} x \neq 0$

$\dpi{100} \implies$ x.(Positive number) $\dpi{100} \neq$ 0.(Positive number)

$\dpi{100} \implies x(x^2 + 4) \neq 0(x^2 + 4)$

$\dpi{100} \implies x(x^2 + 4) \neq 0 \implies x^3 + 4x \neq 0$

Therefore p is false.

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