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Does anyone know of some resources that provide a good introduction to common notation used in formal math? For example, I honestly don't know how to interpret $f: \mathbb{Z} \rightarrow \mathbb{Z}$. I feel like the most difficult time I have with math is not understanding the notation. Thanks!

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Asking for "common notation" is a bit like asking if someone could provide some "common sports trivia" so that you can talk to people about sports at parties. It's going to depend on your background and what you're planning to study. That said, I find it hard to imagine a situation where someone who does not know what $f: \mathbb{Z} \rightarrow \mathbb{Z}$ means would encounter $f: \mathbb{Z} \rightarrow \mathbb{Z}$ in a text, which the person has the appropriate background for, and this notation is not explained in the text. –  Dave L. Renfro May 7 '13 at 19:20
    
Yeah, that makes total sense. I'll just pick up some of the texts I'll be using and read in advance. Now, just for fun, does $f: \mathbb{Z} \rightarrow \mathbb{Z}$ mean something along the lines of "f is a function that maps $\mathbb{Z}$ to $\mathbb{Z}$". I.e., it takes an integer and changes it to another integer? –  Pseudo_Scientist May 7 '13 at 20:30
    
Yes, and more generally, $f:A \rightarrow B$ means $f$ is a function with domain $A$ and "output set" $B.$ This notation gives an "airplane view" of a function in that it only tells you where the input values of the function live, where the output values of the function live, and what symbol is being used to denote the function by (typically $f,$ but it could be $g$ or $H$ or $\phi$ or $\dots$). A standard way to describe a function is illustrated by the following: $f:{\mathbb Z} \rightarrow {\mathbb Z}$ is defined by $f(m) = m^3 - 2m + 5.$ –  Dave L. Renfro May 8 '13 at 19:59
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The Wikipedia page List of mathematical symbols explains the meaning of some basic symbols, with examples of usage. Then there is the category Mathematical notation...

Generally speaking: if the text you are reading uses notation which it does not explain, this means the author assumes the readers are familiar with the notation already. If you are not familiar with it, chances are that you lack the background necessary for reading the book -- which will also manifest itself in ways unrelated to notation.

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