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I was reading a text on Machine Learning, and it defined a loss function as $L:(z,y)\in R\times Y\to L(z,y)\in R$. I know loss function is a measure of how different the predicted outcome is from the actual output, but I couldn't make a sense out of this definition (equation, I guess). I have a hard time understanding the symbols, and then precedence rules that apply to these symbols. I tried searching the web and found a few resources that have definitions that contradict, for example, I read a forward arrow is used to define the domain, and then I read another text where it seemed the arrow was instead used to define the range. Is there any good resource that could introduce me to the symbols used in Mathematics and their general meaning?

Asaf Karagila
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Pawan Subedi
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    $L$ is a function that takes as input a pair $(z,y)$ where $z$ is an element of $\mathbb R$ (which is the set of all real numbers) and $y$ is an element of $Y$. The output of $L$, denoted $L(z,y)$, is an element of $\mathbb R$. – littleO Jun 22 '21 at 19:50
  • @littleO has answered your specific question here. As for a general answer on how you can find out what the different Mathematical symbols mean in literature, I'm actually not too sure of any specific resource that covers everything you'll ever come across. You will end up picking a lot of it up over time. I guess you could always just ask on here also. Sometimes you can also deduce what it means from context. Maybe https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols could be of some help for future reference? – Riemann'sPointyNose Jun 22 '21 at 19:54
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    Although I know what the symbolism intends, it's neither written correctly or sensibility. I get the impression that the author is marginally familiar with mathematical notation and is trying to write to impress instead of inform. To me that string of symbols is like watching a TV show where an actor (not much of a tennis player, if at all) is trying to look like a professional level player (or even just an average U.S. college level player) by wildly swinging their racket, with the camera only showing a close-up of the obviously inappropriate racket swing and not what the ball actually does. – Dave L. Renfro Jun 22 '21 at 20:15

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The author of your text seems to be conflating two things that are best kept separate in describing a function. In this example, we want to state (1) that $L$ maps elements of $R \times Y$ (i.e., pairs $(z, y)$ where $z \in R$ and $y \in Y$) to elements of $R$ and (2) what the value of $L$ is on any given pair $(z, y)$. For (1), we customarily write $L : R \times Y \to R$. For (2), if we have a formula $f(z, y)$ for the value of $L$ on the pair $(z, y)$, we could write $L : (z, y) \mapsto f(z, y)$ to state that.

In the example in your text, no formula is given, so we have the trivial statement that $L$ maps $(z, y)$ to $L(z, y)$, i.e., $(z, y) \mapsto L(z, y)$. Your text has presented a mishmash of (1) and (2), that only makes sense if you already know what the author means. A better mishmash would be to write $L : ((z, y) : R \times Y) \mapsto (L(z, y) : R)$, which you can pick apart as saying that the function $L$ maps a pair $(z, y)$ of type $R \times Y$ to the value $L(z, y)$ of type $R$.

As for resources to help you decipher this kind of thing, textbook references, such as the ones suggested in the other answer are a good start. But be warned that there is no universally agreed formal syntax for mathematics: one author's $\frac{d}{dx}$ is another's $D_x$. Good authors will do their best to explain their notations, but you often have to rely on what is sometimes called mathematical maturity, i.e,. the ability to make well-informed (but, hopefully, testable) guesses about what the signs that you are reading mean.

Rob Arthan
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Discrete mathematics and theoretical calculus/analysis texts have been the most helpful for me, personally. A well-written textbook on either of these topics will provide clear definitions for commonly-used symbols, especially when the symbols relate to sets and functions.

Kenneth Rosen's Discrete Mathematics and its Applications is particularly detailed, and Steven R. Lay's Analysis With an Introduction to Proof is a more brief text which covers some of the same topics.

If you want a quick resource, definitely use Wikipedia's Glossary of Mathematical Symbols. The users have done well to highlight some of the complexities of mathematical notation (authors use different symbols to denote the same idea, or the same symbol to denote different ideas). Additionally, LaTeX character glossaries such as this one can be helpful to learn the name of a particular symbol, but the documentation is not always very in-depth.

I understand why the text you mentioned confused you. This symbol: $\mapsto$ is not used as commonly as $\rightarrow$. The symbols do not mean the same thing, but they are used for similar situations. This answer explains the difference well.

ethan4444444
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