Binary Operations

Definition

Definition

A binary operation \star on a set A is a function \star \colon A \times A \to A.

We commonly denote \star(a,b) as a \star b using infix notation. Binary operations may have the following properties

Definition

Let \star be a binary relation A,

• We call \star associative on A if for all a,b,c \in A

(a \star b) \star c = a \star (b \star c)

• We call \star commutative on A if for all a,b \in A

a \star b = b \star a

Addition is an example of an associative and commutative binary operation on \mathbb{Z}. Multiplication is also an associative and commutative binary operation, whereas something like subtraction is neither associative nor commutative.

(TO-DO Add more information, theorems, examples, etc.)