Sets
An Overview of Sets
A set is a collection of elements. When defining a set, we often use curly braces
. The number of elements in a set is denoted
or
. We can say an element
is a member of the set
by saying
. Note that a set has no repeating elements.
Common sets have their own symbols:
- the set of natural numbers (there is no agreement on whether this contains
or not)
- the set of integers
- the set of rational numbers
- the set of real numbers
- the set of complex numbers
This notation allows us to easily define sets. For example, if we wanted to define a set with all multiple of 3 up to 30, we could do the following:
Assuming we have two sets
and
, we have a way to interact with them.
We say
if both sets have the exact same elements.
The set
contains all elements of interest.
The union of two sets
is a combination set of all elements in
or
.
The intersection of two sets
is a set of all elements that appear in both
and
.
The set-difference of
and
is the set of all elements in
not in
.
The complement of
, denoted
, is all elements of
not in
. You can think of it as
.
If all elements of
are also elements of
, then
is a subset of
and this is denoted
. Note that means it is possible that
. To discount this possibility, we say
is a proper subset of
, denoted
.