Abstract Algebra/Binary Operations

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[edit] 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.

[edit] Associativity

A composition Δ is said to be associative if:

(AΔB)ΔC = AΔ(BΔC)

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

[edit] Commutativity

A composition Δ is said to be commutative if:

AΔB = BΔA

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}

[edit] 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:

eΔx = xΔe = 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 × 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:

eΔf = e
fΔe = f

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

[edit] Ordered Pairs

Most readers should recognize the ordered coordinate pairs from the cartesian coordinate graphing system as an ordered pair of values, (x,y).

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 a1,a2,b1 and b2 respectively:

A = (a1,a2)
B = (b1,b2)

We can see that A = B if and only if

a1 = b1 and a2 = b2


[edit] 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 )

[edit] Answer

operation associative commutative
Addition yes yes
Multiplication yes yes
Subtraction No No
Division No No

The text in its current form is incomplete.

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