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  \[x( x^2 + 1) \neq 0\] 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

\[\neq\]  This is a contradiction.
Hence, q is true.