in the question the contraposition should be contrdiction Sh

in the question the contraposition should be contrdiction

Show, by using the cut axiom and contradiction, that x ^2= 2 has exactly one positive real solution.

4 Problem Show, by using the cut axiom and contraposition, that T2 2 has exactly one positive real solution.

Solution

Suppose x^2=2 has no positive solution

We have x^2-2=0

Its a second degree equation

Comparing to the equation ax^2+bx+c=0,

a=1, b=0, c=-2

Discriminant b^2-4ac=0+4×2=8.

Thus the given equation will have two roots 8^(1/2) and 8^(-1/2),a contradiction to our assumption that there is no positive roots.hence it has one positive root.

Another proof:

By Descartes rule of signs,the no. Of sign changes in the given equation is 1.hence it has a positive root.