Abstract Algebra/Sets and Compositions

A set is a grouping of values, and are generally denoted with upper-case letters. For instance, let's say that A is the set of all first names that start with the letter 'A'. From this definition, we can see that "Andrew" is a member of set A, but "Michael" is not.

Common Sets
Here are some of the common sets:

$$\mathbb{N}$$: The Natural Numbers $$\mathbb{Z}$$: The Integers $$\mathbb{Q}$$: The Rational Numbers $$\mathbb{R}$$: The Real Numbers $$\mathbb{C}$$: The Complex Numbers

The Natural Numbers are the set of non-negative and non-zero integers $$\{1, 2, 3, 4, \ldots\}.$$ The Integers are all the natural numbers, their negative counterparts and zero $$\{\ldots, -2, -1, 0, 1, 2,\ldots\}$$. The Rational numbers are all the numbers that can be formed as a fraction of two integers with a non-zero denominator. The Real numbers include the rational numbers, and also includes all the numbers that cannot be formed as a ratio of two integers. The Complex numbers are all the numbers that involve the imaginary number, i. Notice that C can contain numbers that are imaginary (no real part), real (no imaginary part) and complex (real and imaginary parts).

Set Notation
Frequently, it is required that we define a set by a specific mathematical relationship. For instance, we can say that we want to define the set of all the even integers. Since $$\mathbb{Z}$$ is the set notation for integers, we can say:


 * $$\{x\in \mathbb{Z}: x \mbox{ mod } 2 = 0\}$$

In English, this statement says "All x in set $$\mathbb{Z}$$ such that x modulo 2 equals zero". Or, if we are not familiar with the modulo operation, it is perfectly acceptable to use plain English when defining our set:


 * $$\{x \in \mathbb{Z}: x \mbox{ is even}\}$$

The colon here is read as "such that". This notation will come up a lot in the rest of this book, so it is important for the reader to familiarize themselves with this.

$$a \in A$$ denotes that $$a$$ is an element of A.

Set Operations
A subset S of a set A is a set such that $$s \in S \to s \in A$$. This is denoted as $$S \subset A$$.

The intersection of two sets A and B is the set $$A \cap B = \{s: s \in A \land s \in B\}$$.

The union of two sets A and B is the set $$A \cup B = \{s: s \in A \lor s \in B\}$$.

If $$S \subset A$$, the set $$A-S=\{ s: s \in A \land s \notin S \} $$.

Cartesian Product
A cartesian product between two sets shows the domains of two or more variables. For instance, if we have the variables x and y, and the sets A and B, we can use the cartesian product to show the domains of x and y in terms of A and B:


 * $$A \times B = \{(x, y): x \in A, y \in B\}$$

Compositions
Compositions are operations on a set that act on numbers of the set, and return a value that is in that same set, that is if $$A$$ is a set, a composition is a function $$*:A\times A\to A$$
 * For instance, addition between two integers produces an integer result. Therefore addition is a composition in the integers. Whereas division of integers is an example of an operation that is not a composition, since $$1/2$$ is not an integer.

If we have a set $$A$$, we say that a composition acts on $$A \times A$$ and produces a result in $$A$$. This is also known as closure.

Associativity
A composition Δ is said to be associative if:


 * $$(A \Delta B) \Delta C = A \Delta (B \Delta C)$$

For instance, the addition operation is an associative operation over the integers, Z:


 * $$(1 + 2) + 3 = 6 = 1 + (2 + 3)$$

Notice however, that subtraction is not associative:


 * $$(1 - 2) - 3 = -4,\qquad 1 - (2 - 3) = 2$$

Commutativity
A composition Δ is said to be commutative if:


 * $$A \Delta B = B \Delta A$$

For instance, multiplication is commutative because:


 * $$ 2 \times 3 = 6 = 3 \times 2$$

Notice that division is not commutative:


 * $$ 2 \div 3 = \frac{2}{3}, \qquad 3 \div 2 = \frac{3}{2}$$

Neutral Element
A Neutral Element (or Identity) is an item in E such that a composition in E $$\times$$ E into E returns the other operand. For instance, say that we have a composition Δ, a neutral element $$e \in E$$, and a non-neutral element $$x \in E$$. If Δ is commutative, we have the following relation:


 * $$ e \Delta x = x \Delta e = x$$

For instance, in addition, the neutral element is 0, because 1 + 0 = 1. Also notice that in multiplication, 1 is the neutral element, because 1 &times; 2 = 2.

Each composition may have only one neutral element, if it has any at all. To prove this fact, let's assume a composition Δ with two neutral elements, e and f:


 * $$e \Delta f = e$$


 * $$f \Delta e = f$$

But since e and f are commutative under Δ by definition, we know that e = f.

Ordered Pairs
Ordered pairs are artificial constructions where we set two values into a specific order. More formally, we can define an ordered pair as the set

$$(a,b)=\{\{a\},\{a,b\}\}$$

Let's say that we have two ordered pairs, A and B, comprised of values $$a_1, a_2,  b_1$$  and $$b_2$$ respectively:


 * $$A = (a_1, a_2)$$
 * $$B = (b_1, b_2)$$

We can see that $$A = B$$ if and only if


 * $$ a_1 = b_1 \mbox{ and }  a_2 = b_2$$

Functions
A function is essentially a mapping that connects two values, x and y. We use the following notation to show that our function f is a relationship between x and y:
 * $$(x, y) \in f$$

Notice that x and y form an ordered pair: If we reverse the order of x and y, the relationship will be different (or non-existent). We say that the set of possible values for x is the domain, D, of the function, and the set of possible y values is the Range, R.   In other words, using some of the terms we have discussed already, we say that our function f maps from "D &times; R into R".

Inverses
If f is a function in D &times; R, to R, then f−1 is the inverse of f if it is in R &times; D to D, and the following relationship holds:


 * $$(x, y) \in f, \qquad (y, x) \in f^{-1}$$

Exercise

 * Of the four arithmetic operations, addition, subtraction, multiplication, and division, which are associative? commutative?
 * Using the definition of the ordered pair as a model, give a formal definition for an ordered n-tuple: $$(a_1,a_2,\ldots a_n )$$