I'm practicing and Found this question.
If $ a $ and $b$ are irrational, either prove or disprove that $a + b$ is irrational.
So I tried contradiction (to a + b is irrational).
Let $a$ and $b$ be arbitrary irrational numbers. Assume that$ a + b $is rational.
Then $ a + b = x/y$ for some integers $x$ and $y$.
then $y(a + b) = x$
and $ay + by = x$
Because $x$ was an integer $ay$ is an integer and $by$ is an integer.
then $a$ divides $ay$ and $b$ divides $by$. But that's impossible because a is irrational and b is irrational and y is an integer.
So $a+b$ must be irrational as well.
Now I know this is wrong. Because I found a counterexample as the solution.
$sqrt(2)$ + $-sqrt(2)$ = 0.
Can someone point out my logic mistake? Thank you very much in advance!