Question

Prove by contradiction.

Suppose that x and y are positive integers. Show that the sq root of (x^2 + y^2) is not equal to x + y.

Answer 1

proof by contraction means suppose
\[\sqrt{x^2+y^2}=x+y\]

Answer 2

then arrive at a contradiction,
square both sides
get
\[x^2+y^2=(x+y)^2\]

Answer 3

so
\[x^2+y^2=x^2+y^2+2xy\]
\[0=2xy\] imples either \(x=0\) or \(y=0\)

Answer 4

and since 0 is not a positive integer, this contracts the assumption that both x and y are positive integers