Question

Show that the statement

p : " if x is a real number such that $x^3+x=0$ then x is 0 " is true by

(i) direct method

(ii) method of contrapositive

(iii) method of contradition.

Answer 1

Let q and r be the statement given q : x is a real number such that $x^3+x=0$ r : x is 0.

Then, p : if q, then r.

(i) Direct Method : Let q be true. Then, q is true

$\Rightarrow$ x is a real number such that $x^3+x=0$

$\Rightarrow$ x is a real number such that $x(x^2+1)=0$

$\Rightarrow$ x = 0

$\Rightarrow$ r is true.

Thus, q is true $\Rightarrow$ r is true.

Hence, p is true.

(ii) Method of contra positive : Let r be not true. Then , r is not true.

$\Rightarrow x\ne 0,xϵR$

$\Rightarrow x(x^2+1)\ne 0,xϵR$

$\Rightarrow$ q is not true

Thus, -r = -q.

Hence, p : q $\Rightarrow$ is true.

(iii) Method of contradiction : If possible , let p be not true. Then, p is not true.

$\Rightarrow$ -p is true

$\Rightarrow$ -(p $\Rightarrow$ r) is true.

$\Rightarrow$ q and -r is true

$\Rightarrow$ x is a real number such that $x^3+x=0andx\ne 0$

$\Rightarrow$ x= 0 and $x\ne 0$

This a contradiction.

Hence, p is true.