Hello. I'm Professor Von Schmohawk and welcome to Why U.
This series of lectures is about Algebra.
In these lectures, we will explore the concepts which are fundamental to Algebra.
One of the most fundamental concepts in Algebra is the concept of a set.
A set is simply a collection of objects called elements.
The elements of a set can be anything
numbers, people, letters of the alphabet, other sets, and so on.
Sets are often denoted with capital letters.
Certain important sets are designated with special symbols.
For example, the empty set, which is the set with no members
is designated with a slash "O".
Remember that slash "O" does not represent the number zero.
It denotes the set which contains no elements.
Other important sets are sets of the various number types which we explored in Pre-Algebra.
The set of natural numbers is typically designated by the boldface letter N.
Likewise, the set of integers is designated with the letter Z
the set of rational numbers is designated by Q
and the set of real numbers is designated by R.
Another important set is the set of complex numbers, C
which we will discuss in a lecture later in this series.
We can indicate that an element is a member of a set by using the symbol for set membership.
For instance, we could state that the element one is a member of the set of natural numbers.
We can also indicate that an element is not a member by using the negation of this symbol.
For example, we could state that one-half is not a member of the set of natural numbers.
We define a set by specifying all its members.
There are several ways to do this.
One way is by writing a list of all the members of the set separated by commas
and enclosing the list in curly brackets.
For instance, we can define set A to be a set whose members are the numbers
one, two, and three.
Or we can define set B to be the set whose members are the colors red, white, and blue.
An important rule to remember when listing the members of a set
is that each element must be unique.
No two members of a set may be the same.
For sets with lots of members, if the list of elements continues in some obvious way
an ellipsis can be used to indicate a continuation of the pattern.
For example, the set of integers between three and one-hundred could be written like this.
This can be very useful when specifying the membership of infinite sets.
For instance, the set of natural numbers
can be written with an ellipsis following the first few numbers
to indicate that the list continues forever.
Likewise, the set of integers can be written with an ellipsis on either side
to indicate that the numbers go on forever in both the positive and negative directions.
Fewer or more elements can be listed as long as the pattern is clear.
Instead of listing each member
a set can also be defined by stating the properties that its members must satisfy.
For example, for set A, instead of listing the elements
we could say that A is the set of natural numbers less than four.
And we could define set B as the set of colors of the American flag.
Another option is to enclose the description in curly brackets.
The brackets are read as "the set of all".
For example, A is the set of all natural numbers less than four.
There is also a more rigorous mathematical notation for describing a set
called set-builder notation.
To define a set using set-builder notation
we choose a symbol, typically a small letter, to represent any element of the set.
For example let's use the letter x to represent a typical element of set A.
Now, in the brackets we write x followed by a vertical bar
and then state the properties that every element x must satisfy.
In this case, every element must be a natural number and less than four.
The vertical bar means "such that".
So we would read this definition as
A is the set of all elements, x
such that
x is a natural number and is less than four.
You may sometimes see a colon used instead of a vertical bar.
So far we have seen how to define sets and their members through the use of set notation.
In the next lecture, we will learn about various types of relations between sets.