This document defines common set notation symbols and their meanings. It provides examples of set notation usage and distinguishes between elements of sets and subsets of sets. Specifically, it explains that for something to be a subset it must be defined as a set using brackets, and that while buildings are elements of a university set, the people within buildings are elements of the building sets but not elements or subsets of the university set.

1. Set Notation
Tyler Murphy
February 12, 2014
Symbol
A
|
:
∈
∈
/
∪
∩
⊆
⊂, ,
∅
|S|
Meaning
Denotes the name of the Set
Such that
Such that
Is an element of
Is not an element of
union of two sets (all elements in A or B)
intersection of two sets (elements in A AND B only)
union of multiple sets
intersection of multiple sets
is a subset of
is a proper subset of (not equal to) [all are equivalent notation]
The empty set
The size of S. (The cardinality of S)
Usage
A = {1, 2, 3...}
A = {x | x is even}
A = {x : x is odd}
x∈A
x∈A
/
A∪B
A∩B
∞
n=1 An
∞
n=1 Bn
A⊆B
A⊂B
∅ = {}
S = {1, 2, 3}.|S| = 3.
Also, when thinking about elements of sets versus subsets of sets, consider this. Define
a building as a set with all the people in the building as the elements of that set. Now think
of the university as a set of all the buildings on campus. The elements of the university
are buildings. The subsets of the university are groups of those buildings. However, the
people in the buildings are not elements or subsets of the collection of buildings. They are
elements of the buildings, though.
Consider U = {math building, ILC, Library, Ed Building, SMTC, MECB}.
and MECB = {people in the MECB }.
Note that MECB ∈ U . And {MECB} ⊂ U .
The difference in these statements is subtle but profound. For something to be a subset,
it must be wrapped in set brackets. In this thought line it is clear that MECB ⊂ U is false.
It is an element but not a subset because it is not a set. It is simply an object whereas
{MECB} is a set with one object in it.
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