How to understand the set complement? I need to understand it as I am working with this set on some topology problem.
$$A=\mathbb{R}^2\setminus(\mathbb{Q}\times \mathbb{Q}^c)$$
I can imagine what is going on but is there a proper set theoretic approach to write it.
My efforts
$A$ is the complement of points like $$\{(a,b)\;|\;a\in \mathbb{Q}, b\in \mathbb{Q}^c\}$$
So complement should look some thing like $\{(a,b)\;|\;a\in \mathbb{Q}^c, b\in \mathbb{R}\}\cup \{(a,b)\;|\;a\in \mathbb{R}, b\in \mathbb{Q}\}$
What you wrote is correct and can be written more simply as
$$A=(\mathbb Q^C\times \mathbb R)\cup (\mathbb R\times\mathbb Q)$$
To prove the two sets are equal, you can easily prove that
$$\mathbb{R}^2\setminus(\mathbb{Q}\times \mathbb{Q}^c)\subseteq (\mathbb Q^C\times \mathbb R)\cup (\mathbb R\times\mathbb Q)$$ and $$(\mathbb Q^C\times \mathbb R)\cup (\mathbb R\times\mathbb Q)\subseteq \mathbb{R}^2\setminus(\mathbb{Q}\times \mathbb{Q}^c)$$ are both true statements.