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
. We can say an element
is a member of the set
. 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
- 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
, we have a way to interact with them.
if both sets have the exact same elements.
contains all elements of interest.
The union of two sets
is a combination set of all elements in
The intersection of two sets
is a set of all elements that appear in both
The set-difference of
is the set of all elements in
The complement of
, is all elements of
. You can think of it as
If all elements of
are also elements of
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