Operations

Why

We have seen several concepts that consist of associating a pair of sets with a third set. For example, set unions and set intersections

Definition

An operation (or binary operation, law of composition) on a set $A$ is a function from $A \times A$ to $A$.

Roughly speaking, operations combine (or compose) elements. We operate on ordered pairs.

Example: set operations

Let $X$ be a set. Define $g: \powerset{X} \times \powerset{X} \to \powerset{X}$ by $g(A, B) = A \cup B$. Then $g$, the function which associates with two sets their union is an operation on $\powerset{X}$. Likewise, define $h : \powerset{X} \times \powerset{X} \to \powerset{X}$ by $h(A, B) = A \cap B$.

Naming their properties

$\cup$ has several nice properties. For one $A \cup B = B \cup A$ and $(A \cup B) \cup C = A \cup (B \cup C)$.

An operation with the first property, that the ordered pair $(A, B)$ and $(B, A)$ have the same result is called commutative. An operation with the second property, that when given three objects the order in which we operate does not matter is called associative. $\cap $ shares these properties with $\cup$.

We call the operation of forming unions the function $(A, B) \mapsto A \cup B$. We call the operation of forming intersections the function $(A, B) \mapsto A \cap B$. We call the operation of forming symmetric differences the function $(A, B) \mapsto A \symdiff B$. Since forming unions commutes and is associative and likewise with forming intersections, forming symmetric differences also commutes.

Algebras

Of course, any operation is defined on some set. For this reason, we define an algebra (or algebraic structure) as an ordered pair whose first element is a non-empty set and whose second element is an operation on that set. The ground set (or underlying set, carrier set, domain) of the algebra is the set on which the operation is defined.