Consider the statement : p : If x a real number such that `x^3 +4x =0 ` then x=0 prove that p is a true statement using :
(i) direct method d
(ii) method of contradiction
(iii) method of contrapositive

Explanation

(i) Dircet Method :
Let `x^3 +4x =0` where ` x in R` .Then
`x^3 +4x=0 rArr x(x^2+4)=0`
`rArr x=0 [ because x^2 +4 ne 0 " for x " in R]`
(ii) Method of contradiction: If possible `let x^3 +4x =0` and `x ne 0` Then , `x(x^2+4)=0 and x ne 0, rArr x^2+4 -0`
But , this is a contradiction arises by assuming that
`x^2+4 =0 " and" x ne 0 rArr x^2 +4 ne R`
Since the contradiction arises by assuming that
`x^2+4 =0 and x ne 0 rArr 0 ` is a ture statement.
(iii) Method of contrapositive :
We have to prove that `x^3+4x=0 rArrx=0`
Let `p : x^3+4x=0 amd q : x =0`.
We shall prove that `~ q rArr ~ p`.
Let `x ne 0`. Then, `x^3+4x=x(x^2+4) ne 0 [ :' x ne 0 and x^2+4 ne0]`.
Thus, `~ q rArr ~p" and therefore", p rArr q`.