Question

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 contradition.

Answer 1

p : "If x is a real number such that$x^3+x=0$then x is 0".
Let q and r be the statements.
Here,
q: x is a real number such that $x^3+x=0$.

r: x is 0.

(i) Direct method
Let q be true.
To obtain $x^3+x=0$we have:
^{$x(x^2+1)=0$}

or, x = 0
Thus, r is true.
Hence, "if q, then r" is a true statement.

(ii) Method of contrapositive
Let r not be true.
r is not 0.

If $x(x^2+1)\ne 0$, then q is not true.
Hence, "if ~q, then ~r" is a true statement.

(iii) Method of contradiction
Let q not be true.
Then,
$~$q is true
$~$(q $\Rightarrow$r) is true.

q &$~$r is true


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

Then, x is not 0.

x = 0 and x$\ne$0
This is a contradiction.
Hence, q is true.