Introduction to Sets
- Forget everything you know about numbers.
- In fact, forget you even know what a number is.
- This is where mathematics starts.
- Instead of math with numbers, we will now think about math with "things".
What
is set?
Definition
What is a set? Well, simply put, it's a
collection.
First we specify a common property among
"things" (this word will be defined later) and then we gather up all
the "things" that have this common property
For example, the items
you wear: shoes, socks, hat, shirt, pants, and so on.
I'm sure you could come up with at least a hundred.
This is
known as a set.
Or
another example is types of
fingers.
This
set includes index, middle, ring, and pinky.
So it is just things grouped
together with a certain property in common.
Notation
There is a fairly simple notation for sets. We simply
list each element (or "member") separated by a comma, and then put
some curly brackets around the whole thing:
The curly brackets { } are sometimes called
"set brackets" or "braces".
This
is the notation for the two previous examples:
{socks, shoes, watches, shirts, ...}
{index, middle, ring, pinky}
{index, middle, ring, pinky}
Notice how the first example has the "..."
(three dots together).
The three dots ... are called an
ellipsis, and mean "continue on".
So that means the first
example continues on ... for infinity.
(OK, there isn't really an infinite amount of things you could
wear, but I'm not entirely sure about that! After an hour of thinking of
different things, I'm still not sure. So let's just say it is infinite for this
example.)
So:
·
The
first set {socks, shoes,
watches, shirts, ...} we call
an infinite set,
·
the
second set {index, middle,
ring, pinky} we call a finite set.
But sometimes the "..." can be used in the middle
to save writing long lists:
Example:
the set of letters:
{a, b, c, ..., x, y, z}
In this case it is a finite set (there are
only 26 letters, right?)
Numerical
Sets
So what does this have to do
with mathematics? When we define a set, all we have to specify is a common
characteristic. Who says we can't do so with numbers?
Set of even numbers:
{..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}
And the list goes on. We can come up with all different
types of sets.
There can also be
sets of numbers that have no common property, they are just defined that way.
For example:
For example:
{2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}
Are all sets that I just randomly banged on my keyboard to produce.
Why
are Sets Important?
Sets are the fundamental
property of mathematics. Now as a word of warning, sets, by themselves, seem
pretty pointless. But it's only when we apply sets in different situations do
they become the powerful building block of mathematics that they are.
Math can get amazingly complicated
quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis,
Linear Algebra, Number Theory, and the list goes on. But there is one thing
that all of these share in common: Sets.
Universal
Set
At the start we used the word
"things" in quotes. We call this the universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to our question.
|
Then our sets included integers. The universal
set for that would be all the integers. In fact, when doing Number Theory, this
is almost always what the universal set is, as Number Theory is simply the
study of integers.
However in Calculus (also known as real analysis), the
universal set is almost always the real numbers.
And in complex analysis, you guessed it, the universal set is the complex
numbers
Notation
When talking about sets, it is fairly standard to use Capital Letters to
represent the set, and lowercase letters to represent an element in that set.
So for example, A is a set, and a is an element in A. Same with B and b, and C and c.
So for example, A is a set, and a is an element in A. Same with B and b, and C and c.
Now you don't have to listen
to the standard, you can use something like m to represent a set without breaking
any mathematical laws (watch out, you can get π years in math jail for dividing by 0),
but this notation is pretty nice and easy to follow, so why not?
Also, when we say an element a is in a set A, we use the symbol
to show it.
And if something is not in a set use
.
And if something is not in a set use
Example:
Set A is {1,2,3}. We can see that 1
A, but 5
A
Equality
Two sets are equal if they
have precisely the same members. Now, at first glance they may not seem equal,
so we may have to examine them closely!
Example:
Are A and B equal where:
A is the set whose members are the first four positive
whole numbers
B = {4, 2, 1, 3}
Let's check. They both contain 1. They both contain 2.
And 3, And 4. And we have checked every element of both sets, so: Yes,
they are equal!
And the equals sign (=) is
used to show equality, so we write:
A =B
Subsets
When we define a set, if we
take pieces of that set, we can form what is called a subset.
So for example, we have the set {1, 2, 3, 4, 5}. A subset of this is {1, 2, 3}. Another subset
is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it
contains an element (6) which is not in the parent set. In general:
A is a subset of B if and only if every element of A is in B.
So let's use this
definition in some examples.
Is a subset of B, where A = {1, 3, 4} and B = {1, 4,
3, 2}?
- 1 is in A, and 1 is in B as well. So far so good.
- 3 is in A and 3 is also in B.
- 4 is in A, and 4 is in B.
That's all the elements of A, and every single one is
in B, so we're done.
Yes, A is a subset of B
Note that 2 is in B, but 2 is not in A. But remember,
that doesn't matter, we only look at the elements in A.
Let's
try a harder example.
Example:
Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B?
And is B a subset of A?
Well, we can't check every element in these sets,
because they have an infinite number of elements.
So we need to get an idea of
what the elements look like in each, and then compare them.
The sets are:
A = {..., -8, -4, 0, 4, 8, ...}
B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}
By pairing off members of the two sets, we can see
that every member of A is also a member of B, but every member of B is not a
member of A:
A is a subset of B, but B is not a subset of A.
Proper
Subsets
If we look at the definition
of subsets and let our mind wander a bit, we come to a weird conclusion.
Let A be a set. Is every element in an element in A? (Yes, I wrote that
correctly.)
Well, umm, yes of course, right?
So doesn't that mean that A
is a subset of A?
This doesn't seem very proper,
does it? We want our subsets to be proper.
So we introduce (what else but) proper
subsets.
A is a proper subset of B if and only if every
element in A is also in B, and there exists at
least one element in B that
is not in A.
This little piece at the end is only there to make sure
that A is not a proper subset of itself. Otherwise, a proper subset is exactly
the same as a normal subset.
Example:
{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper
subset of {1, 2, 3}.
Example:
{1, 2, 3} is a proper subset of
{1, 2, 3, 4} because the element 4 is not in the first set.
Notice that if A is a proper
subset of B, then it is also a subset of B.
Even
More Notation
When we say that A is a
subset of B, we write A
B.
Or we can say that A is not a subset of B by A
B ("A is
not a subset of B")
When we talk about proper subsets, we take out the line
underneath and so it becomes A
B or if we want
to say the opposite, A
B.
Empty
(or Null) Set
This is probably the
weirdest thing about sets.
As an example, think of the
set of piano keys on a guitar.
"But wait!" you say,
"There are no piano keys on a guitar!"
And right you are. It is a set with no elements.
This is known as the Empty
Set (or Null Set). There
aren't any elements in it. Not one. Zero.
It is represented by:
Or by {} (a set with no elements)
Some other examples of the empty set are the set of countries south of the
south pole.
So what's so weird about the empty set? Well, that part
comes next.
Empty
Set and Subsets
So let's go back to our
definition of subsets. We have a set A. We won't define it any more than that,
it could be any set. Is the
empty set a subset of A?
Going back to our definition of subsets, if every element in the empty set
is also in A, then the empty set is a subset of A. But what if we have no elements?
It takes an introduction to logic to understand this, but
this statement is one that is "vacuously" or "trivially"
true.
A good way to think about it is: we can't find any elements in the
empty set that aren't in A, so it must be that all elements in the empty
set are in A.
So the answer to the posed question is a resounding yes.
The
empty set is a subset of every set, including the empty set itself.
Order
No, not the order of the
elements. In sets it does not
matter what order the elements are in.
Example:
{1,2,3,4} is the same set as {3,1,4,2}
When we say "order" in sets we mean the
size of the set.
Just as there are finite and
infinite sets, each has finite and infinite order.
For finite sets, we represent the order by a number, the
number of elements.
Example: {10, 20, 30, 40} has an order of 4.
For infinite sets, all we
can say is that the order is infinite. Oddly enough, we can say with sets that
some infinities are larger than others, but this is a more advanced topic in
sets.
QUESTIONS?
