Sets: Terminologies, Notations, and Operations
As a preparation for more posts on probability, statistics, permutations and combinations, we familiarized ourselves last week with the different terminologies and notations of probability. We continue in this post by studying set terminologies, notations, and operations. Note that this is also the third post in the Set Primer Series; the first and second are Introduction to Sets and Subset: a set contained in a set.
Universal Set
The universal set is the set that contains all the elements under discussion. If we talk about the letters in the English alphabet, then the universal set contains all the 26 letters. In set theory, universal set is usually denoted by .
In the following discussion, we let be the set of integers,
be the set of even integers, and
be the set of odd integers. The following are the common operations on sets.
Intersection
If sets and
have elements in common they form a set written as
. This is the intersection of
and
.
Example: If we let and
then
.
Union
The union of sets ,
, is the set
containing all the elements of
and
or both.
Example: If we let and
then
.
Example: If we let be the set of even integers and
be the set of odd integers, then
.
Disjoint
If and
have no elements in common, they are said to be disjoint.
Example: The sets and
mentioned above have no elements in common so they are disjoint sets. In notation, we use the symbol
to denote an empty set. Therefore,
.
Example: From above, it is also clear that
Difference
is the set consisting of all elements of
which are not in
.
Example: .
Complement
The set of elements in which are not in
is the complement of
denoted by
.
Example: If we let be the set of integers,
be the set of odd integers, then
.
Image Source: Wikipedia