A binary operation is a function that maps ordered pairs of elements of a set to elements of the set. For example, addition of natural numbers maps every pair of natural numbers to their sum, so addition is a binary operation on natural numbers.
Apart from the common operations such as addition, multiplication, dot-product, etc., a binary operation is commonly denoted by placing an asterisk between the elements: \(a*b\).
If a binary operation has the property that \(a*b=b*a\) for every \(a\) and \(b\) in the set, then the operation is said to be commutative. If a binary operation has the property that \((a*b)*c=a*(b*c)\) for all \(a\), \(b\), and \(c\) in the set, then the operation is said to be associative.