How do we express a straight line of $R^2$ in set notation?

Question

How do we express a straight line of $R^2$, with equation $y=x$, in set notation? Is this okay?

$ \{y \ | \ y=x, x \in R \} $

Answer 1

You want to have a set of points in the plane. A point in the plane is usually written as $(x, y)$. We start with that: $$ \{(x, y)\in \Bbb R^2\mid{} $$ (The $\in \Bbb R^2$ is to make it clear to anyone reading this line that we are actually talking about points in the plane, and implicitly that $x$ and $y$ are real numbers.)

Now, which $x$ and $y$ pairs are we after? It's all the pairs where $x = y$ and no other pairs. So we add that, and we're done: $$ \{(x, y)\in \Bbb R^2\mid x = y\} $$

Answer 2

The appropriate notation would be $\{(x,y)\in\Bbb R^2\,:\,y=x\}$. An alternative could be $\{(t,t)\,:\,t\in\Bbb R\}$, though it would not be too nice to use when it is not clear from context that it is suppesed to be a subset of $\Bbb R^2$.