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I am trying to do some self-studying and am consistently getting bogged down by sophisticated mathematical notation. For example, something simple like this in a book:

Let $X$ be a non empty set and then we define a metric $d$ on X as the function:

$$ d: X \times X \rightarrow \mathscr R $$

Here, I have a confusion as to how I should read it. So d is a function that takes the quantity $X \times X$ and maps it to a real number? I am not sure why this is not written as

$$ d:X \rightarrow \mathscr R $$

I find this notation really in the first equation in the online book:

https://www.math.ucdavis.edu/~hunter/book/ch1.pdf

Caleb Stanford
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Luca
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1 Answers1

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The notation means that $d$ is a function that takes a pair of numbers and maps them to the reals. The pair of numbers could be thought of as an ordered pair $(x,y)$ where $x \in X$ and $y \in X$, and the tuple $(x,y)$ is in the Cartesian product of $X$ with itself: $X \times X$.

Emily
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    To clear it up a little more, it may be more familiar to use a well-known function. Addition of integers is a function $ + : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$. However, the function $f(x) = x+5$ is just a function of one variable, so we have $f : \mathbb{Z} \to \mathbb{Z}$. – Emily Jan 05 '15 at 21:21
  • This clears up things a lot. I was seeing this notation when I was trying to read a bit about Lie groups as well (associated with the multiplication between group elements) and could not figure out what it really meant. Thanks a lot! – Luca Jan 05 '15 at 21:45