Welcome to an introduction
to sets and set notation.
A set is a collection of objects
whose contents can be clearly determined.
Elements are the objects in a set.
Objects will typically be mathematical
which are either elements or not elements of a set,
similar to a statement, which is true or false.
Much of discrete mathematics is the study of finite sets.
Capital letters are generally used to name sets
and a set is usually represented in one of three ways.
Number one, a list, all elements
of a set are listed between curly brackets.
If a set is infinite, we list as many elements
as needed to reveal a pattern.
Below, set A contains all of the even integers.
Notice the dot, dot, dot to the left and right
which indicates the pattern continues both
to the left and the right, resulting in an infinite set.
Number two, set notation.
This is a formal mathematical notation that defines
the properties that an element of a set must satisfy.
For example, the set A of all even integers
can be expressed as follows.
Set A is a set containing x
which is an in integer such that x is even.
And because x has to be an integer,
we can also say set A is a set of all integers
such that x is even.
Number three, a Venn diagram.
This is a graphical representation of a set
that can be useful to identify relationships
among different sets.
For example, in the Venn diagram below,
we have set A and set B.
The overlapping region of set A and set B
would contain all the elements that are in set A and set B.
And now's talk about some special sets.
First, the symbol on the left indicates the empty set.
The empty set is a set that contains no elements.
We can also denote the empty set
by using curly brackets with nothing between them.
Capital U is for the universal set.
This is a set of all elements in a particular context
that we are concerned about.
Capital N is used for the natural numbers
which is the set containing zero,
one, two, three, four, and so on.
Next, capital Z is the set of integers.
Q is the set of rational numbers, which are the numbers
that can be expressed as a ratio of integers.
Capital R is used for the set of real numbers
which are all the numbers on a number line.
P of A is the power set of A,
which is a set of all subsets of A.
And we'll talk more about subsets in just a moment.
Now let's talk about the different notation used for sets.
To begin, as we stated earlier,
we use braces or curly brackets
to enclose the elements of a set
and the elements are separated by commas.
So here we have the set containing one, two and three.
We use the colon to represent such that.
Here we have an example of set notation
where this set is a set of all x
such that x is greater than two.
We made to say this is a set of all numbers
such that x is greater than two.
We use the next symbol that looks like an E
to show an element is in a set.
Here we have two is an element
of the set containing one, two, and three.
If we put a slash through the same symbol,
this indicates something is not an element of a set.
So here we have four is not an element
of the set containing one, two, and three
because the element four is not in the set.
The next symbol looks like a rounded inequality symbol.
This is used for a subset.
We read this as A is a subset of B
which indicates every element of A
is also an element of B
and for a subset, set A can equal set B.
Next, we have the notation for a proper subset.
We say A is a proper subset of B
which indicates every element of A is also an element of B
but for a proper subset, set A cannot equal set B.
An easy way to remember the difference of subset
and proper subset is to think of these as inequalities.
The notation for subset looks like less than or equal to.
In this case, the two sets can be equal.
And the notation for proper subset looks like less than.
In this case, set A cannot equal set B.
The next symbol looks like an upside-down capital U
which is used for intersection.
We read this as the intersection of A and B
or just A intersect B.
This is the set containing all elements
which are elements of both A and B.
This is analogous to conjunctions.
The next symbol looks like a capital U.
This is used for union.
We say the union of A and B, or A union B.
This is a set containing all elements
which are elements of A or B or both.
This is analogous to disjunctions.
The next symbol, which looks like a multiplication symbol,
is the Cartesian product of A and B.
This is a set of all ordered pairs,
little a comma little b,
where little a is an element of set A
and little b is an element of set B.
Next we have two common symbols for a set difference.
We read this as A minus B.
This is the set containing all the elements of A
which are not elements of B.
Next, we have a couple common symbols
for the compliment of A.
One is a bar over A and another is a capital C
in what looks like the exponent position.
The compliment of A is a set of everything
which is not an element of A.
And then finally, we also have a couple symbols
for cardinality of A.
The cardinality of A is the number
of distinct elements in A.
And now let's talk about some special considerations
when using set notation.
First, the set containing zero
is not equal to the empty set,
since the set containing zero contains one element,
the element zero, and the empty set contains no elements.
The number zero also doesn't equal the empty set
since zero is a number and the empty set is a set.
And the empty set is a subset of every set.
For d, notice we have this strange set
where we have four elements.
One element is one, another element is b.
The third element is a set containing x and y,
and the fourth element is the empty set.
x is not an element of this set
since x is not one of the four elements.
Notice a third element is the set containing x and y.
So we could say that the set containing x and y
is an element of this odd set.
For e, the set containing one is not an element
in the set containing one, two, and three
since the set containing one is a set, not an element.
So we can say one is an element
of the set containing one, two, and three.
So it's important to recognize the set containing one
is not the same as the element one.
For f, three is not a subset of the set
containing one two and three
since three is an element and not a set.
A subset must be a set.
So we could say the set containing three is a subset
of the set containing one, two, and three.
And then for g, we have another strange set here
where the set has four elements.
One element is the number one.
Another element is the set containing one and two
and then we have the set containing three,
and then finally we have the empty set.
So we can say the set containing three is an element
of the set because one of the elements
is the set containing three.
But the set containing three is not a subset of this set
because the set containing three is an element in this set.
We would have to say the set
containing this set containing three
is a subset of this set,
which is kind of strange and normally doesn't occur,
but that would be correct.
And part h is related.
We can say the set containing three is a subset of this set
because this set does contain the element three.
Again, the element three is not the same
as a set containing three.
It is correct to say that three is an element of this set.
So notice a difference here.
We can say the set containing three is a subset of this set
and we can say three is an element of this set.
And then for i, we have another example
of set builder notation where
the set containing zero, two, four, six, and so on,
which should be the even natural numbers,
can be expressed as a set of x
where x is a natural number such that x is even
or we can just say the set of all natural numbers
such that x is even.
I hope you found this helpful.