Introduction to Abstract Algebra/Lecture 1

Introduction to the Course
The study of abstract algebra has numerous applications to fields outside of mathematics, such as chemistry and in particular, physics. The primary objective of this introductory course in Abstract Algebra is to aquaint you with studying Group Theory. Groups are one of the most fundamental algebraic structures.

There are other fundamental structures, such as fields and rings. Those will be covered in a future course in algebra. For now, we'll start off with a brief review of set theory. For this, I'll assume you've already seen some logical symbols. Problem set exercises will be assigned following the lecture.

Elements
We'll start off with some set theory. A set is simply a collection of objects; be it a set of books, the complete 'set' of Star Wars movies, a set of baseball cards, whatever. So let's suppose $$A$$ is a set. We say $$a \in  A$$ when the element $$a$$ belongs to $$A$$. We say $$a \notin A$$ when $$a$$ does not belong to $$A$$.

A few very important characteristics about sets:


 * The elements of a set are unique. That means the set $$\left\{1, 2, 3\right\}$$ and $$\left\{1, 2, 3, 3\right\}$$ describe the exact same set.
 * A set has no order. That means the set $$\left\{1, 2, 3\right\}$$ and $$\left\{2, 3, 1\right\}$$ describe the exact same set.

Okay? Lets move on.

Subsets
Definition: Let $$A$$, $$B$$ be sets. We say A is a subset of B, and write $$A \subseteq B$$, if and only if $$ a \in A \Rightarrow a \in B$$.

This means that every element in A is also contained in B. Now we'll see what it means for two sets to be equal.

Definition: Let $$A$$, $$B$$ be sets. Then we say $$A$$ and $$B$$ are equal, and write $$A=B$$, if and only if $$A \subseteq B$$ and $$B \subseteq A$$.

We see here that two sets are equal if every element in A is contained in B and every element in B is contained in A. This definition of set equality invokes some interesting results:

Example: Let $$A = \left\{1,2,3\right\}$$. Then, $$A \subseteq A$$, i.e. $$A$$ is a subset of $$A$$. (Try reason it using our definition of subsets above).

As seen in the above example, a set is necessarily a subset of itself, which does not correspond to our conventional meaning of the word subset that implies a "smaller" set. We therefore invent a new definition for our conventional meaning of subsets.

Definition: Let $$A$$, $$B$$ be sets. We say that $$A$$ is a strict subset of $$B$$, and write $$A \subset B$$, if and only if $$A \subseteq B$$ and $$A \neq B$$.

Now for some set operations.

Set operations
Here are some basic set operations:


 * 1) $$A \cup B = \left\{x \mid x \in A \text{ or } x \in B \right\}$$. This is called the set union.
 * 2) $$A \cap B = \left\{x \mid x \in A \text{ and } x \in B\right\}$$. This is called the set intersection.
 * 3) $$A \setminus B = \left\{x \mid x \in A \text{ and } x \notin B\right\}$$. This is called the set difference.
 * 4) $$A \times B = \left\{(a,b) \mid a \in A \text{ and } b \in B\right\}$$. This is called the Cartesian product.

If you need further review on sets and their operations, please refer to Wikibooks textbook on Set Theory. We now turn our attention to a big concept in all of mathematics, relations. Relations will play a large part in our study of Algebra later on.

Equivalence Relations
Definition: A binary relation on a set, A, is a subset, R, of A x A.  Alternatively, we can say that a subset of A x A is a relation on A.

Here is some notation. Notation: a ~ b if $$(a,b) \in R$$. In other words, a is related to b if both a and b belong to R.

Here is a simple example. Let's say that we have a set of students in a classroom. Then two students are related if they sit in the same row of seats.

Definition: A relation ~ is called an equivalence relation if the following hold:

1): a ~ a $$\forall a \in A$$. This is called reflexivity. 2): If a ~ b $$\Rightarrow$$ b ~ a.  This is called symmetry. 3): If a ~ b and b ~ c, $$\Rightarrow$$ a ~ c. This is called transitivity.

Lets go back to our previous example of students in a classroom. Lets say we have a set of students in a classroom. Then a student is related to his/herself. If student a and student b are sitting in the same row, then student b and student a are in the same row. Finally, if student a and student b are sitting in the same row, and student b and student c are in the same row, then student a and student c are in the same row. Therefore, it is an equivalence relation. Lets look at another example.

Example: Let $$(a,b) \in \mathbb{Z}$$. Show a ~ b if 2 | (a - b). That is, 2 divides (a-b).

Therefore, ~ is an equivalence relation.
 * 1) If a ~ a, then 2 | (a - a).  So ~ is reflexive.
 * 2) If a ~ b $$\Rightarrow$$ 2 | (a - b), then 2 | (b - a) $$\Rightarrow$$ b ~ a.  So ~ is symmetric.
 * 3) If a ~ b and b ~ c, $$\Rightarrow$$ 2 | (a - b) and 2 | (b - c). $$\Rightarrow$$ 2 | [(a - b) + (b - c)].  That is, 2 | (a - c).  $$\Rightarrow$$ a ~ c. So ~ is transitive.

Now we look at another important idea, the equivalence class.

Equivalence Classes
Definition: Let ~ be an equivalence relation on a set A and let $$a \in A$$. Then the equivalence class of a is defined to be: cl(a) = {$$b \in A |$$ a ~ b}.

Going back to our previous example, there is an equivalence relation if two students are in the same row. Then the equivalence class is that row of students.

Example: If a ~ b with 2 | (a - b), what is the equivalence class of zero?

Well, cl(0) = {$$b \in A |$$ 0 ~ b} = $$cl(a) =$$ {$$b \in A |$$ 2 | (0 - b)} = $$cl(a) =$$ {$$b \in A |$$ 2 | b} = $$2 \mathbb{Z}$$ (the set of all even numbers.) So cl(0) = $$2 \mathbb{Z}$$. What then is cl(1)? Going through the same steps, cl(1) = $$2 \mathbb{Z} + 1$$

Next Time
That's all for this lecture. Next time we'll tie up the loose ends in set theory and begin our discussion on mappings. Please see the associated problem set for exercises.