Proof Writing · Hard problem

Assume $x$ is rational and $y$ is irrational and $x + y$ is rational, and derive a contradiction.

Assume $x + y$ is rational and show $x + y$ is irrational.

Assume $x$ is irrational or $y$ is rational and $x + y$ is irrational, and derive a contradiction.

Assume $x + y$ is irrational and show $x$ is rational and $y$ is irrational.