Abstract Algebra/Binary Operations

Binary Operations

A binary operation (say ◦) on a set A is a function ◦ : A × A → A

Let a,b be two elements in A, then a ◦ b ∈ A

In the case of this operation, two elements are in the act, so the adjective binary is in use.

Compositions

Compositions are operations that act on a 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:

(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:

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

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:

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

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 a + 0 = a for all a. Also notice that in multiplication, 1 is the neutral element, because a × 1 = a for all a.

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.

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

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: $(a_1,a_2,\ldots a_n )$

Answer

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

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