$ \{(b\overline{a},1);\;(a,b) \in \mathbb{C}^2\}=\mathbb{C}\times \{1\} $

Question

Let us consider the following subset of $\mathbb{C}^2$

$$ S=\{(b\overline{a},1);\;(a,b) \in \mathbb{C}^2\}. $$

I want to prove that $S=\mathbb{C}\times \{1\}$. We have $S\subset\mathbb{C}\times \{1\}$

Answer 1

Take $a=1$ and choose any $b\in\mathbb{C}$ then $b\overline{a}=b$ is an arbitrary complex number. Hence we have that $\mathbb{C}\times\{1\}\subseteq S$. Also we require $b\overline{a}$ to be complex as $\mathbb{C}$ is closed under multiplication. Hence $S\subseteq\mathbb{C}\times\{1\}$. These two facts imply that $S=\mathbb{C}\times\{1\}$.