Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lecture we used Venn Diagrams to illustrate set operations such as intersections
and unions
as well as relations between sets such as subsets and supersets.
In this lecture, we will see how Venn Diagrams can be used
to illustrate the creation of a set's "complement" and "relative complement".
The complement of a set is the collection of all elements
which are not members of that set.
If we have some set, A
then the complement of set A
is the set of all elements not contained in A.
However, there is one problem with this definition.
It may not always be clear what we mean by "all" elements.
For example, if you order a hotdog with everything except onions
you assume that the vendor knows that "everything" refers to all the condiments.
But if you and the vendor don't agree on the definition of the word "everything"
this could create big problems.
So when we say that the complement of a set
is the collection of all elements which are not members of that set
we must first agree on what is meant by "all elements".
The set of all elements is called the "universal set" or the "universe"
and is typically designated by the capital letter "U".
We can define set U to be anything we like
as long as all the members of the set whose complement we are taking
are elements of U.
The complement of set A will then be all the elements of U which are not members of A.
As an example on the primitive island of Cocoloco
we could define the universal set to be the set of all human inhabitants.
Then if set A contains all male Cocoloconians
the complement of set A would be the collection of all female Cocoloconians.
We indicate the complement of a set
by writing the set's name with a bar over it.
You may also see the complement denoted by the set's name with a superscript, c
or a prime symbol.
As another example, let's say that the universal set
is the set of all integers
which, as you may recall, is designated by the letter Z.
The set of natural numbers is contained in this universal set
so the complement of the set of natural numbers
is the set of all integers which are not natural numbers
in other words, all the negative integers plus the number zero.
Using set-builder notation, we can define a set's complement formally as
"The complement of set A
is the set of all elements, x
such that
x is a member of a universal set, U
and x is not a member of A.
In cases where it is obvious which set is being used as the universal set
it may be omitted from the set definition.
Or sometimes you may see the statement that
every element is a member of the universal set
written to the left of the vertical bar.
This is read as
"The complement of set A
is the set of all elements, x which are members of the universal set, U
such that
x is not a member of A.
The result we get when we take the complement of a set
will depend on how we define the universal set.
For instance, we showed that if the universal set is the set of integers
then the complement of the set of natural numbers
is the set of integers that are not natural numbers
in other words, all the negative integers plus the number zero.
However, if we define the universal set to be the set of real numbers
then the complement of the set of natural numbers
is the set of real numbers that are not natural numbers
in other words, the entire number line excluding the points which represent positive integers.
Forming a set's complement is a unary set operation
in the same way that squaring a number is a unary numerical operation.
Unary operations involve only one operand.
The result of this set operation is another set
just as the result of this numerical operation is another number.
An operation which is closely related to the forming of a set's complement
is forming a set's "relative complement".
We saw that to form the complement of any set, A
where A is a subset of some universal set, U
we form A's complement by excluding the elements in U which are members of A.
The remaining elements form A's complement.
Now let's say that instead of the universal set
we use some other set, B
which doesn't necessarily contain A but may only intersect A.
When we exclude all elements from B which are also members of A
we call the remaining elements the "relative complement of A in B".
Since the relative complement of A in B
is the collection of all elements in B with the elements of A removed
the relative complement is sometimes written as "B minus A".
You may also see this denoted as "B back-slash A".
In this context "back-slash A" means "not A"
in other words, all elements in B which are not in A.
So the complement and relative complement operations are actually quite similar.
The only difference is that in the complement operation
set A must always be completely contained in the other set
which we call the universal set
where the relative complement operation
does not require the other set to contain A.
The complement operation is sometimes called the "absolute complement"
so as not to be confused with the relative complement.
We now have several operations which create new sets by combining sets in various ways.
In the next lecture, we will introduce one additional set operation
called the "symmetric difference".