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Abstract Algebra

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Set Notation, Element of a Set

ABSALG-KLCG2V

In mathematics, we usually use set notation to describe properties satisfied by the elements of a set and by those elements only.

The elements of the set, or their description, are given inside curly brackets $\{\;\;\}$:

e.g. $\{7,8,9,10\}$ defines a set by a list and this same set is described by $\{x\in{\mathbb Z}\mid\, 7\le x\le 10\}$,

where $\mid$ means such that, and ${\mathbb Z}$ is the set of all integers. (Often $:$ is used instead of $\mid$ for such that).

We have $7\not\in S$, for $S$ given by:

A

$S=\{$number of days in the week$\}$

B

$S=\{x\in{\mathbb Z}\mid\,x=3k+2,\; {\rm for\;some\;integer}\;k\}$

C

$S=\{x\in{\mathbb Z}\mid\,x=3k+4, \;{\rm for\;some\;integer}\;k\}$

D

$S=\{x\in{\mathbb Z}\mid\, x\;{\rm odd},\;x<10\}$

E

$S=\{{\rm integers\;that\;divide}\;21\}$