SETS

Sets

Sets is a collection.

Notation

Simple notation for sets. List each element or member separated by a comma, and then put some curly brackets around the whole thing like :

The brackets { } are sometimes called "Set brackets"

The three dots ... are called an ellipsis and "continue on".

Numerical Sets

When we define a sets, all we have to specify is a common characteristic.

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}

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 defined that way.

For example:

{2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}

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. Equality 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} They both contain 1, 2, 3 and 4. We have checked every element of both sets, so: Yes, They are equal. Subsets When we define a set, if we take pieces of that set, we can form what is called subset. 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. Some examples: Is A 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 remember, that doesn't matter, we only look at the elements in A.