Discrete Math 1: Sets

Students will eventually google their teachers, this is a fact of life. Therefore I should take the opportunity of being googled to provide help to students in my classes so I will start writing about discrete math now that I figured out how to draw better pictures.

Sets are nice because anybody can understand the with minimal effort. This is mainly because the language of basic set theory is familiar to us in every day life. A set is just a collection of things that we call elements.

Some examples of sets would be;
the set of all humans (the elements of the set are humans)
the set of all men (the elements of the set are men)
the set of all women (the elements of the set are women)
the set of all chairs (the elements of the set are chairs)
the set of all people named Brandon (the elements of the set are Brandons)
the set of numbers 1 and 2 (the elements of the set are the numbers 1,2)
the set of all integers (the elements of the set are integers)
the set of all real numbers solving the equation $x+1=0$ (the only element is -1)
and pretty much anything you can imagine. Each individual set is rarely important, the important thing to study is properties of sets.

The set is the entire collection, while an element of the set is something in the set. This is very important to understand (thus the bolding is justified).

The idea of sets instantly brings about the idea of subsets. A subset of a set, is also a set, but is contained in the original set. A more precise way to state this is that for any two sets A and B, A is a subset of B if every element of A is an element of B and we use the notation
$A\subset B$

For example, the set of all women is contained in the set of all humans. The set of all men is also contained in the set of all humans. These are both subsets of the set of all humans.

A small annoying detail to mention is that any set is a subset of itself by definition. The above notation is for when we know that A is a strict subset of B, with no possibility of being equal to B.
If A might have a chance of being the exact same set as B, but is definitely a subset at the very least, then we use the notation
$A\subseteq B$

We also have operations on sets, just like we have with numbers. When talking about some sets, we can look at their union or intersection (and some other stuff later).

Math people like to use drawings to represent things. For sets, we like to use Venn Diagrams to help us visualize properties of sets. This will help us understand the definition of intersection, union, and subset in a better way.

For example here are two sets A and B, represented by circles.

With a bit of color, we can see the intersection of sets.

Where $A\cap B$ is just the notation for A intersect B. Of course where the two circles meet is where the intersection is. The union of A and B is given by both circles put together and is denoted $A\cup B$. An example with some actual sets is the following;

This picture can easily be extended to three sets using circles, but it doesn’t work so well with four circles or more. With up to three sets, you can easily look at all of the separate intersections and unions.

$A = \{1,2,3,4\}$
$B = \{3,4,5,6\}$
Then we have that
$A\cap B= \{3,4\}$
$A\cup B= \{1,2,3,4,5,6\}$

Notice that I don’t write 3 and 4 twice just because they are both in A and also in B. When I union sets, I only write the overlapping elements once. This is because the definition of a set only depends on exactly what elements are in the set. So if something is in a set once, we just say it is in the set and there is no need to write it multiple times.

Next we have a picture representing subsets. In this, the circle representing the set B is completely contained in the circle representing the set A.

That’s all for now.