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I have a statement as such:

$\mathbb{Z_+} \triangle E = \{ x \in \mathbb{Z_+} : x \space \% \space 2 \neq 0 \space and \space x \in \mathbb{Z} : x < 0 \space and \space x \space \% \space 2 = 0 \}$

($E$ represents all even numbers, positive and negative. Not important to the actual question, but it is important to the statement above.)

My question is: is there any way I can get those "and"s out of my statement? They seem so out of place. Is there any notation for "and"?

Bonus: Is my statement correct?

Jaken
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    What does the second : mean? What do you mean by "all even numbers, even and odd"? – littleO Aug 29 '15 at 05:47
  • @littleO typo! Meant to say all even numbers positive and negative. – Jaken Aug 29 '15 at 05:48
  • @Jaken: Unless your goal is to make your mathematics as incomprehensible and annoying as possible, please don't write this sequence of symbols. Common language is far better for communicating to other humans, and is not in any way "less formal" or "less mathematical". I'd far prefer $${\text{positive integers}}\mathrel{\triangle}{\text{even integers}} = {\text{odd positive integers and even non-positive integers}}$$ – Zev Chonoles Aug 29 '15 at 05:51
  • @ZevChonoles I am currently taking a discrete math class and it's one of the first courses where mathematical notation is introduced. Our professor has been giving us sentences to try and change from text to math notation just to see if we understand what the symbols mean. I don't want to make math annoying, I just don't have a choice. – Jaken Aug 29 '15 at 05:54
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    People tend to use comma when giving out conditions such as $\forall x, x \in \mathbb{Z}$ – Fraïssé Aug 29 '15 at 05:55
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    The second colon is not standard when writing sets. You could express the right hand side as a union of two sets. – littleO Aug 29 '15 at 06:18

2 Answers2

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There's the usual &. And there's: $\land$, but this is only used in propositional logic; so to use that you should make your statement too formal in shape in order for it not to look weird. It might confuse some as well because it may mean different things in different situations.

Also, if you have been doing advanced math for too long, "," might magically become a symbol for "and", though the same symbol is sometimes used as "or" and "such that" as well; it depends on the context (and can be understood what to mean from there).

"Is my statement true?" Yes.

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Assuming that $\mathbb{Z}_{+}$ has the standard meaning of $$\mathbb{Z}_{+}=\{\text{positive integers}\}=\{x\in\mathbb{Z}:x>0\}$$ then your answer is not quite right, since $$\begin{align*} \mathbb{Z}_{+}\mathrel{\triangle}E&=\{x\in\mathbb{Z}_{+}:x\notin E\}\cup\{x\in E:x\notin\mathbb{Z}_{+}\}\\\\ &=\{x\in\mathbb{Z}_{+}:x\not\equiv0\bmod 2\}\cup\{x\in\mathbb{Z}:x\in E\wedge x\notin\mathbb{Z}_{+}\}\\\\ &=\{x\in\mathbb{Z}_{+}:x\not\equiv0\bmod 2\}\cup\{x\in\mathbb{Z}:x\equiv 0\bmod 2\wedge x\not>0\}\\\\ &=\{x\in\mathbb{Z}_{+}:x\not\equiv0\bmod 2\}\cup\{x\in\mathbb{Z}:x\equiv 0\bmod 2\wedge x\leq 0\} \end{align*}$$ The error is that "not positive" doesn't mean "negative". Your set misses the element $0$.

Zev Chonoles
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