What is a set? Simply put, a set is a collection of objects. Its' members are called elements. For example, call A the set of all animals. Some elements of A might be cows, chickens, and sheep. The elements of a set are usually contained in curly brackets, like so:
To say that a cow is an element of A, we would write
If one set has all the elements of another set, we call the smaller set a subset. For example, a subset of A could be a set called M, the set of all mammals. To say that M is a subset of A, we would write
.Part Two: Operations on Sets.
Union "adds" all the elements of a set together. For example, take two sets
Then the union of B and C is denoted
See what happened there? B is in union with C, forming a new set that "adds" all their elements together.
Intersection is what two sets have "in common". For instance, take two sets
The intersection of B and C is then written as
The new set formed is made up of elements that belong to both B and C.
The set-theoretic difference, also known as the relative complement of two sets, takes one set and "subtracts" the elements of another set. For example, take two sets
Then the relative complement of B and C (in that order!) is denoted
All we did here is remove the elements of C from B. Simple, huh?
Part Three: Special Sets.
Before we get into the exciting stuff, we need to introduce some very special sets:
The empty set is the set with no elements. The symbol for the empty set is Ø.
A rational number is a number that can be expressed as the ratio of two integers. The set of all rational numbers is denoted as Q.
An irrational number is any number that cannot be expressed as a ratio of two integers.
A real number is any number that is not complex. You can think of real numbers as the union of rational and irrational numbers. The set of real numbers is denoted as R.
Part Four: Infinite Sets and Countability.
Now for the fun stuff. whee
An infinite set is, intuitively speaking, a set that "goes on forever". An infinite set has an infinite number of elements.
Can one infinite set be bigger than another? The concept of countability addresses this issue.
Definition: A set is called "countable" if there can be a one-to-one correspondance established between the set in question and the set of positive integers, Z+.
What does this mean? Well, think about the term "one-to-one correspondence". All this means is that for every element of Z+, there is an element to "pair up with" in the set in question.
Now we can establish a few things about countable sets. I will not provide proofs due to space issues, but if you would like to see them, PM me:
Theorem One: Every subset of a countable set is countable.
Theorem Two: The union of a countable set is itself countable.
Theorem Three: Every infinite set has a countable subset.
We need to define another important concept now: the concept of equivalence. Two finite sets are called equivalent if they have the same number of elements. What of infinite sets? All countably infinite sets are equivalent to Z+, because they have the same number of elements.
Now for the question of uncountability:
Definition: A set is said to be "uncountable" if we cannot establish a one-to-one correspondence between the set in question and Z+.
We have defined two types of infinity: uncountability and countability. Intuitively speaking, uncountable sets are "bigger" than countable ones.
How does this work? Think about it: they really do represent two different concepts! For example, the set of all points on the number line is uncountable; there is an infinite amount of numbers between the integers. Uncountability represents this concept, while countability represents the more common "going on forever" concept of infinity.
I hope I have done a good job explaining these fascinating concepts -- feedback, questions, and discussion are very welcome. 3nodding
