Intuitively, it seems to me that $|\mathbb{R}^{2}| > |\mathbb{R}|$. My intuition says something to the effect of: There exists a bijection between $\mathbb{R}$ and $\{(x,0) : x \in \mathbb{R}\}$, and the second set is a subset of $\mathbb{R}^{2}$.
But I remember reading somewhere that $|\mathbb{R}^{2}| > |\mathbb{R}|$ is not true, though I may be mistaken.
I know only the very basics about countable sets which are infinite. I also know that $\mathbb{R}$ is uncountable.
To be clear, I am looking for:
- Is $|\mathbb{R}^{2}| > |\mathbb{R}|$?
- Why?
- Any quick resource/lecture notes to quickly brush up on related concepts.
The most relevant question I could find was: Is it proper to say that two infinite sets are the "same size" if there is a bijection between them?