# Subtraction of elements in sets

Let A be a set of finite elements.

$A=\{1,2,3,4,5\}$

If I want to remove one element and show I removed one element, how should I do?

Pseudo mathematical notation:

$A - \{2\} = \{1,3,4,5\}$

Thank you very much!

n

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7
People often prefer to write $A\setminus\{2\}$ or $A \smallsetminus \{2\}$ to simply $A -\{2\}$ but what you wrote is fine. See here for example where `\setminus` $\setminus$ is used. – t.b. Sep 18 '11 at 17:45

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Why is your notation psuedo-mathematical? – Srivatsan Sep 18 '11 at 17:45

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i thought it wasn't mathematical enough to be mathematical – graphtheory92 Sep 18 '11 at 18:21
Somehow, the previous comments sound like a Zen koan. – Ilmari Karonen Sep 18 '11 at 18:48

## 1 Answer

Your notation above is actually used in set theory.

In general if you have two sets $A$ and $B$, the difference $A - B$ is the set

$A - B = \{x \in A : x \notin B\}$

Also, note that $\{1, 2, 3, 4, 5\} - \{2, 6\} = \{1,3,4,5\}$. $B$ need not be a subset of $A$.

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One reason to prefer $A \setminus B$ is that I've also seen $A - B$ occasionally used for the set $\{a-b: a \in A, b \in B\}$. That is, by that definition $\{1,2,3,4,5\} - \{2,6\} = \{-5,-4,-3,-2,-1,0,1,2,3\}$. Of course, there's no problem as long as you explain which definition you're using, but $A \setminus B$ avoids that ambiguity completely. – Ilmari Karonen Sep 18 '11 at 18:55
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