Hello. I'm Professor Von Schmohawk
and welcome to Why U.
In the previous lecture we saw how to define sets
and their members
through the use of set-builder notation.
In this lecture, we will learn about several types of relations between sets.
One such relation is "equality".
If two sets contain exactly the same elements they are said to be equal.
For example, set A, which consists of the elements one, two and three
and set B consisting of the elements two, three, and one
are equal since they both contain the same elements.
Notice that it doesn't matter that their elements are listed in different orders.
This is because the order in which the elements of a set are listed is irrelevant.
The set consisting of one, two, and three
is exactly the same as the set consisting of two, three, and one.
The list does not imply that there is any particular order to the elements.
It only says that every element listed is a member of the set.
So if sets A and B contain the same elements then they are equal.
Another important relation between sets is the relation of "subset".
If all the elements of set A are also contained in set B
then we say that set A is a subset of set B.
This relation is denoted using the subset symbol.
If two sets, A and B, are equal then they are subsets of each other.
This is because all the elements of A are members of B
and all the elements of B are also members of A.
So any two sets that are equal are subsets of each other.
And since every set is equal to itself every set is also a subset of itself.
So the subset symbol can be used
whenever all the elements of A are also members of B
whether A has the same number of elements as B or has fewer elements.
But when A has fewer elements, we can be more specific and call A a proper subset of B.
We denote this using the proper subset symbol
which is the subset symbol without the line underneath.
In addition to the relation of subset, a set can also be a "superset".
In any case where set A is a subset of set B
we can also say that set B is a superset of set A.
This relation is denoted using the superset symbol, which is the subset symbol reversed.
Likewise, if A is a proper subset of B then B is a proper superset of A.
But what about the empty set, the set containing no elements?
The empty set is considered to be a subset of every other set
including itself.
In fact, the empty set is the only set which is a subset of every set.
So far we have represented sets by listing their members
or using set-builder notation
or by drawing little ovals with the elements inside.
In the next lecture, we will see how to visualize sets using Venn Diagrams.