for all real numbers x and y, if x - Prove proposition?

Question

Prove the following proposition:


1- for all real numbers x and y, if x is rational and y is irrational, then x+y is irrational.

Answer 1

If a number, x, is rational, then there exists some pair of integers, say a & b, such that x = a/b.

Suppose x+y is rational, then we can find two more integers,
say c & d, such that x+y = c/d.

So we have a/b + y = c/d

y = c/d - a/b

y = (bc - ad) / (bd)

but (bc-ad) and (bd) are both integers, so y must be rational

Therefore x+y cannot be rational if x is rational and y is irrational

Answer 2

x+y is rational if x+y = a/b (b not = 0)
So a = bx +by
Since y is irrational then bx + by is irrational --> a is irrational.
Therefore x+y is irrational.

Answer 3

think x+y is rational. Then x+y-y = x is rational, contradiction. That shows that our supposition became incorrect so x+y is irrational. I used the certainty that the style of two rational numbers is a rational extensive style.