Question

Use contrapositive to prove this. Let x, y ∈ R. If x and y are both positive, then (√ x + y) does not equal √x + √y.

Answer 1

See explanation

Explanation:

Let x, y ∈ R.

If x and y are both positive, then
`√(x+y)\\eq √(x)+√(y)`

Suppose that

`√(x+y)=√(x)+√(y)`

Square both sides of this equation:

`(√(x+y))^2=(√(x)+√(y))^2\\ \\x+y=(√(x))^2+2√(x)√(y)+(√(y))^2\\ \\x+y=x+2√(x)√(y)+y\\ \\2√(x)√(y)=0`

Then

`√(x)=0\ \text{or}\ √(y)=0\\ \\x=0\ \text{or}\ y=0`

But x and y are both positive, so the assumption is false