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I'm teaching myself math and I can't figure out what the below symbols mean in a textbook:

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I did figure out that $\Bbb Z$ = integer set and $\Bbb N$ = natural, and $\Bbb R$ = real. But what do all the pluses and dots and $\{0\}$ mean? Eg how is the first $\Bbb Z$ different from the second?

Broader question - what book / tutorial exists that can help me get comfortable with notation?

Thanks a ton.

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  • $\begingroup$ Every book has its own notation, and while most of it is conventional, any good book also explains any notation they use. Are you sure that these aren't explained wherever you have these from? $\endgroup$ – Arthur Oct 31 '19 at 9:18
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    $\begingroup$ $(\mathbb Z, +)$ is the structure of integers with the only operation of addition while $(\mathbb Z, \cdot)$ is the structure of integers with the only operation of multiplication. $\endgroup$ – Mauro ALLEGRANZA Oct 31 '19 at 9:18
  • $\begingroup$ $(\mathbb Z,+)$ is the ordered pair whose first element is the set of integers and whose second element is the addition operation on the integers. $(\mathbb Z, \cdot)$ is the ordered pair whose first element is the set of integers and whose second element is the multiplication operation on the integers. $(\mathbb R^n,+)$ is the ordered pair whose first element is $\mathbb R^n$ (that is, the set of all ordered $n$-tuples of real numbers) and whose second element is the addition operation on $\mathbb R^n$. $\endgroup$ – littleO Oct 31 '19 at 9:21
  • $\begingroup$ The + and · refer to the algebraic operation on the set: addition or multiplication. For instance $(\Bbb Z,+)$ means one considers the set of integers, endowed with addition. $\endgroup$ – Bernard Oct 31 '19 at 9:21
  • $\begingroup$ Notation varies from author to author. You'll pick up the standard notation over time, and many books have an index listing the notation they use. For your question, a bit of context would be useful, but it looks like your book is considering algebraic structures. For example, the notation $(\mathbb{Z}, +)$ is commonly used to indicate that the integers are considered as a group with the group operation being addition. The notation $(\mathbb{Z},\cdot)$ is less common, since the integers don't form a group with multiplication, but it could be a monoid. That's why context would be useful. $\endgroup$ – EuYu Oct 31 '19 at 9:25
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  • $(\mathbb Z,+)$ is the set $\mathbb Z$ endowed with the operation addition;
  • $(\mathbb Z,\cdot)$ is the set $\mathbb Z$ endowed with the operation multiplication;
  • $(\mathbb N_0,+)$ is the set $\mathbb N_0$ endowed with the operation addition;
  • $\mathbb R\setminus\{0\}=\{x\in\mathbb R\mid x\neq0\}$;
  • $(\mathbb R^n,+)$ is the set $\mathbb R^n$ endowed with the operation addition;
  • $(\mathbb Z^n,+)$ is the set $\mathbb Z^n$ endowed with the operation addition.
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    $\begingroup$ Also note that $X^n$ in this context means the cartesian product (en.wikipedia.org/wiki/Cartesian_product) of $n$ copies of $X$ $\endgroup$ – David Oct 31 '19 at 9:20
  • $\begingroup$ Some people denote $\Bbb N_0$ by $\Bbb N$, i.e., the set $\Bbb Z_{\ge 0}$. $\endgroup$ – Dietrich Burde Oct 31 '19 at 9:33
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The symbol represent

  • $(\mathbb Z,+)$ a group on integer numbers by sum
  • $(\mathbb Z,\cdot)$ a group on integer numbers by product
  • $(\mathbb N,+)$ a structure on natural numbers by sum
  • $\mathbb R \setminus \{0\}$ is the set of real numbers without the zero element (indeed the symbol "$\setminus$" indicates the subtraction for sets)
  • $n\in \mathbb N$ means that $n$ belongs to the set of natural numbers
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  • $\begingroup$ @J.W.Tanner Yes of course! I fix that $\endgroup$ – user Oct 31 '19 at 9:37
  • $\begingroup$ Careful, $(\Bbb N,+)$ is not a group, because $n\ge 1$ has no additive inverse. $\endgroup$ – Dietrich Burde Oct 31 '19 at 10:07
  • $\begingroup$ @DietrichBurde You are right! I'll use structure for that. $\endgroup$ – user Oct 31 '19 at 10:14

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