Fun Video: Complement of a Set Video Lecture | Mathematics (Maths) Class 11 - Commerce

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".