Binary Operations

A binary operation is a rule that combines two elements of a set to produce another element. Usually denoted by symbols like *, ⊕, ⊗, or Δ, it is a formal way of generalizing arithmetic operations like addition and multiplication.

1. Basic Definition

If S is a non-empty set, a binary operation * on S is a function that assigns to each ordered pair (a, b) of elements of S, a unique element a * b in S.

2. Properties of Binary Operations

• Closure: A set S is closed under an operation * if for every a, b ∈ S, the result a * b is also in S.
• Commutativity: An operation * is commutative if a * b = b * a for all a, b in the set.
• Associativity: An operation * is associative if (a * b) * c = a * (b * c) for all a, b, c in the set.
• Distributivity: If we have two operations (* and ⊕), * is distributive over ⊕ if a * (b ⊕ c) = (a * b) ⊕ (a * c).

3. Identity Element (e)

An element e is called the identity element with respect to the operation * if for every element a in the set:
a * e = e * a = a

Example: In addition of real numbers, 0 is the identity because a + 0 = a.

4. Inverse Element (a⁻¹)

If e is the identity element, then an element x is the inverse of a if:
a * x = x * a = e

Example: In addition, the inverse of 5 is -5 because 5 + (-5) = 0 (the identity).

5. Solved Example

Question: An operation is defined by a * b = a + b + 2. Find the identity element.
Solution: Let e be the identity.
a * e = a
a + e + 2 = a
e + 2 = 0
e = -2.
The identity element is -2.