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The notation of "$*$" started being used in my proof textbook in the section of equivalence relations and partitions yet it never once said what it means.

An example from the textbook:

Let $\mathbb{Z}^* = \mathbb{Z} - \{0\}.$ Define the relation on $\mathbb{Z} \times \mathbb{Z^*}$ by, for all $a,c \in \mathbb{Z}$ and all $b,d \in \mathbb{Z}$

What does "$*$" mean?

David
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    The first sentence in the quote gives you the answer. "Let $\mathbb Z^=\mathbb Z-{0}$". This means that we define* $\mathbb Z^*$ to be $\mathbb Z-{0}$. – Wojowu Apr 05 '19 at 19:41
  • It's just a decoration which distinguishes the decorated symbol from the undecorated one.There are many similar notations: $X^*$, $X'$, $\hat{X}$, $\tilde{X}$, $\bar{X}$, etc – MPW Apr 05 '19 at 19:41
  • Exactly what it is equal to, of course: it is $;\Bbb Z;$ without zero. It is very usual in many contexts. – DonAntonio Apr 05 '19 at 19:41
  • The $$ is there just to distinguish $\mathbb Z^$ from $\mathbb Z$. The definition of $\mathbb Z^*$ is exactly $\mathbb Z-{0}.$ – D. Brogan Apr 05 '19 at 19:41
  • Let $\mathbb{Z}^* = \mathbb{Z} - {0}.$ –  Apr 05 '19 at 19:48
  • Thank you for the comments, I was thinking that $\mathbb {Z^}$ meant something other then just $\mathbb Z^=\mathbb Z-{0}$ – David Apr 05 '19 at 19:50

2 Answers2

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In an algebraic context, many authors use $A^*$ to denote the set $A$ without the zero element. In your specific example, the author uses $\mathbb Z^*$ to denote the set $\mathbb Z$ without $0$, i.e. $\mathbb Z-\{0\}$. So, $*$ is just used in a context of notation and does not denote any particular operation.

In a similar manner, it also common to write $\mathbb R^*$ for $\mathbb R-\{0\}$, $\mathbb C^*$ for $\mathbb C-\{0\}$, etc.

blub
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The notation refers to all non-zero elementes, in your case, of the integers $\mathbb Z$. As it is defined by excluding $0$ from the integers by $\mathbb Z^*=\mathbb Z-\{0\}=\mathbb Z\setminus\{0\}$.

mrtaurho
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