Binary Operations

A binary operation is a rule that combines any two elements from a set to produce a single element, typically within the same set. You already use binary operations every day: addition takes two numbers and returns a number, multiplication does the same, and even set intersection takes two sets and returns a set. What makes this topic exciting in algebra is that mathematicians — and exam writers — can invent entirely new binary operations using any rule they like, as long as the rule is well-defined. In the context of Relations & Functions, a binary operation is formally treated as a function of two inputs. If the set is S, then a binary operation ★ on S is a function ★ : S × S → S, meaning it maps every ordered pair (a, b) from the Cartesian product S × S to exactly one element of S. This requirement — that the output always stays inside S — is called closure, and it is the first property we check for any operation. Custom binary operations appear frequently in high school mathematics competitions, standardised tests, and university entrance exams. A problem might define a ★ b = 2a − b + ab and then ask you to evaluate expressions, check for commutativity, or find identity elements. Mastering the definitions, properties, and systematic approach to these problems builds the algebraic thinking you will need for calculus, abstract algebra, and beyond.

Key takeaways
  • Closure
  • Commutativity
  • Associativity
  • Identity Element
  • Inverse Element

Closure

An operation ★ on set S is closed if for every a, b ∈ S, the result a ★ b is also in S. Example: addition on the set of integers ℤ is closed because adding two integers always gives an integer. However, division on ℤ is NOT closed because 1 ÷ 2 = 0.5 ∉ ℤ. Always verify closure first by checking whether the formula can produce outputs outside S.

Commutativity

An operation ★ is commutative if a ★ b = b ★ a for all a, b ∈ S. To prove commutativity, show the two expressions are always equal. To disprove it, find a single counterexample. For instance, if a ★ b = 2a − b, then 3 ★ 5 = 2(3) − 5 = 1 but 5 ★ 3 = 2(5) − 3 = 7. Since 1 ≠ 7, the operation is not commutative.

Associativity

An operation ★ is associative if (a ★ b) ★ c = a ★ (b ★ c) for all a, b, c ∈ S. Standard addition and multiplication are associative. Subtraction is not: (5 − 3) − 2 = 0, but 5 − (3 − 2) = 4. For a custom operation, substitute carefully and simplify both sides to check if they are always equal.

Identity Element

An element e ∈ S is an identity for ★ if a ★ e = e ★ a = a for every a ∈ S. For addition on ℝ, the identity is 0. For multiplication, it is 1. To find the identity for a custom operation a ★ b = a + b − ab, set a ★ e = a: a + e − ae = a, so e − ae = 0, giving e(1 − a) = 0 for all a. This only works if e = 0. Check: a ★ 0 = a + 0 − 0 = a ✓. So e = 0 is the identity.

Inverse Element

For each element a ∈ S, an inverse a' is an element satisfying a ★ a' = e (and a' ★ a = e), where e is the identity. For the operation a ★ b = a + b − ab with identity e = 0, find a': a + a' − a·a' = 0, so a'(1 − a) = −a, giving a' = a/(a − 1) for a ≠ 1. This means every element except 1 has an inverse under this operation.

The Key Equation

a⋆b=f(a, b),where f:S×S→Sa \star b = f(a,\, b), \quad \text{where } f: S \times S \to S
  • \star — Binary operation symbol (—): A placeholder symbol for any custom-defined binary operation rule.
  • a,\, b — Operands (—): Any two elements chosen from the set S; order may matter if the operation is non-commutative.
  • S \times S — Cartesian product (—): The set of all ordered pairs (a, b) with a, b ∈ S; this is the domain of the operation.
  • f(a,b) — Rule / formula (—): The explicit algebraic expression defining how the two inputs are combined to produce the output.

Worked Example

Problem. A binary operation ★ is defined on ℝ by a ★ b = a² + b − 3ab. (i) Find 2 ★ 4. (ii) Is ★ commutative? Justify your answer. (iii) Find the value of x such that x ★ 2 = 6.

x=3±13x = 3 \pm \sqrt{13}

Answer. x = 3 + √13 or x = 3 − √13

Figure
Diagram
Binary Operation as a Function: S × S → SS × S(1, 2)(3, 5)(2, 2)(4, 1)(0, 7)S−116211−35a ★ b = a² + b − 3abEach pair maps to exactly one output in S
Origins · 1 minGeorge Boole and how this was discovered
GB
George Boole
1815–1864 · British

George Boole created Boolean algebra, a system of binary operations (AND, OR, NOT) on the set {0, 1} that forms the mathematical foundation of all modern computing and digital logic circuits. His 1854 book 'An Investigation of the Laws of Thought' formalised the idea that logical reasoning could be expressed as algebraic operations on sets — a revolutionary idea that unified logic and mathematics. Boole was entirely self-taught in mathematics. He taught himself Latin, Greek, and advanced mathematics from library books and never attended university, yet he became the first Professor of Mathematics at Queen's College Cork (now University College Cork) in Ireland.

  • Domain and Range of Functions — A binary operation is a function from S × S to S; understanding domain (all ordered pairs) and range (possible outputs) is essential for verifying closure and the well-definedness of the operation.
  • Cartesian Product — The domain of a binary operation is the Cartesian product S × S, so fluency with ordered pairs and Cartesian products is a prerequisite for this topic.
  • Introduction to Groups — A group is a set equipped with a binary operation satisfying closure, associativity, identity, and inverse — the four properties studied here become the axioms of an algebraic group.
  • Composition of Functions — Function composition is itself a binary operation on the set of functions; studying it deepens understanding of how operations combine two objects into one.