Binary Operations

A binary operation is a rule that combines two elements from a given set, resulting in another element from the same set.
If S is a set, and * is a binary operation, then if we take any two elements a and b from the set, a * b also belongs to the set S.
Example: Addition (+) and multiplication (x) are binary operations on the set of integers.

Properties of Binary Operations

Closure Property: This states that a set is closed under a binary operation if the operation on any two elements of the set always produces an element in the same set.
Associativity: A binary operation * is said to be associative if, for any elements a, b, and c in set S, (a * b) * c = a * (b * c).
Identity Element: An element e in set S is called an identity for the binary operation * if for every element a in S, e * a = a * e = a.
Inverse Element: For each element a in our set S, there is another element b in the same set such that the binary operation * on a and b results in the identity element, i.e. a * b = b * a = e.

Addition (+) and multiplication (x) on the set of real or complex numbers are common binary operations.
Subtraction (-) and division (/) are operations that do not always satisfy the properties of binary operations.
Concatenation is a binary operation on a set of strings or sequences.
Set union ∪ and intersection ∩ are binary operations on a set of sets.

Example: The set of natural numbers under the operation of addition is a binary operation as it satisfies all the properties.
Non-Example: The subtraction operation on the set of natural numbers is not a binary operation as it doesn’t always result in a natural number (negative values, for instance).