For all real number x and y, if x is rational and y is irrational, then x+y is irrational?

Question

Prove proposition:
For all real number x and y, if x is rational and y is irrational, then x+y is irrational

Answer 1

If x+y were rational then y= (x+y)-x would be rational, which it is not.
Hence, x+y is irrational. Happy?

Answer 2

Assume that your proposition is false and show that it leads to a contradiction.

Assume that x+y is rational. This means that x+y = a/b for some a and b which are integers.

Now show that this implies that y = c/d for some integers c and d. This is the bit you have to do yourself.

This means that y is rational. But we know it isn't, so we have a contradiction, and the original proposition is proved.

Answer 3

proof by contradiction
let x+y=p p is rational
then y=p-x
p,x are rational then p-x should be rational which implies y is rational which is not true