An idea involving Cartesian products

Question

While reading Analysis-I by Terence Tao, I came across the notion of tuples, which are objects of a Cartesian product. In one example he writes that the empty Cartesian product is not {}, but a singleton set {()}, in which () is described is the 0-tuple. I didn’t understand how this could be true.

Answer 1

You could probably take it as a definition. For some intuition as to why it should have one element, you could think about the fact that $|X\times Y|=|X|\times|Y|$ when $X$ and $Y$ are finite sets. If $E$ is the empty Cartesian product, then we should have $X\times E=X$ (where I will be a little vague about what exactly I mean by "$=$"!). In particular, $|X|\times|E|=|X\times E|=|X|$, so $|E|=1$.