Binary operations, closure and commutativity – Binary Operations (Senior 4 Mathematics) | Ganzaa

Binary Operations

Lessons

Senior 4 · Mathematics

Binary Operations

4 lessons

Lesson 1 of 4

Binary operations, closure and commutativity

25 min

Arithmetic on a clock face

A journey starts at 9 o'clock and lasts 5 hours. On a 12-hour clock it ends at 2 o'clock, so on the clock "9 plus 5 is 2". This addition takes two hours from the set $\{1, 2, \ldots, 12\}$ and always returns an hour from the same set: you can never "fall off" the clock face. Ordinary addition of whole numbers, subtraction of integers, the union of two sets and the clock's addition all share this pattern: two elements go in, one element of the same set comes out. Such a rule is called a binary operation, and this unit studies the structures it creates.

Definition

Binary operation: a binary operation $*$ on a non-empty set $E$ is a rule that assigns to every ordered pair $(a, b)$ of elements of $E$ exactly one element $a * b$ of $E$ . In other words, it is a mapping $E \times E \to E$ , $(a, b) \mapsto a * b$ .

Closure: $E$ is closed under an operation $*$ when $a * b \in E$ for all $a, b \in E$ . An operation is a binary operation on $E$ exactly when it is defined for every pair and $E$ is closed under it.

Notation: symbols such as $*, \circ, \oplus, \otimes, \triangle$ are used; $(E, *)$ means "the set $E$ with the operation $*$ ".

Example: $+$ is a binary operation on $\mathbb{N} = \{0, 1, 2, \ldots\}$ ; subtraction is not, because $2 - 5 = -3 \notin \mathbb{N}$ .

The set matters

The same rule can be a binary operation on one set and fail on another. (Here $\mathbb{Q}^*$ means the non-zero rational numbers and $\mathbb{R}_+^*$ the positive real numbers.)

Rule	Set	Binary operation?	Reason
$a - b$	$\mathbb{N}$	No	$2 - 5 = -3 \notin \mathbb{N}$
$a - b$	$\mathbb{Z}$	Yes	the difference of two integers is an integer
$a \div b$	$\mathbb{Z}$	No	$1 \div 2 \notin \mathbb{Z}$ , and $a \div 0$ is undefined
$a \div b$	$\mathbb{Q}^*$	Yes	the quotient of two non-zero rationals is a non-zero rational
$\sqrt{ab}$	$\mathbb{N}^*$	No	$\sqrt{2 \times 3} = \sqrt{6} \notin \mathbb{N}^*$
$\sqrt{ab}$	$\mathbb{R}_+^*$	Yes	$ab \gt 0$ , so $\sqrt{ab}$ is a positive real
$A \cup B$ , $A \cap B$	subsets of a set $S$	Yes	a union or intersection of subsets of $S$ is a subset of $S$

Warning

Two ways to fail. A rule is not a binary operation on $E$ if it ever gives an answer outside $E$ (not closed) or if it is undefined for some pair (such as $a \div 0$ ). To prove that a rule fails, one counterexample is enough. To prove that it works, you need an argument for all $a$ and $b$ , written with letters.

Worked example 1: is it a binary operation?

Decide, with reasons: (a) $a * b = a + b - a b$ on $\mathbb{Z}$ (b) $a \circ b = \frac{a}{b}$ on $\mathbb{Z}^*$ (c) $a \oplus b = a^2 + b^2$ on the set $O$ of odd integers (d) $a \,\triangle\, b = |a - b|$ on $\mathbb{N}$ .

Solution

Yes. Sums, products and differences of integers are integers, so $a + b - ab \in \mathbb{Z}$ for all $a, b \in \mathbb{Z}$ .
No. $1 \circ 2 = \frac{1}{2} \notin \mathbb{Z}^*$ .
No. $1 \oplus 1 = 2$ , which is even. (In fact, odd + odd is always even, so $O$ is never closed.)
Yes. $a - b$ is an integer and its absolute value is a non-negative integer, so $|a - b| \in \mathbb{N}$ .

Worked example 2: calculating with a new operation

On $\mathbb{Z}$ , define $a * b = 2 a - b + 3$ . Find $2 * 5$ , $5 * 2$ , $(2 * 5) * 1$ and $2 * (5 * 1)$ .

Solution Replace $a$ by the left element and $b$ by the right one.

$2 * 5 = 4 - 5 + 3 = 2$ and $5 * 2 = 10 - 2 + 3 = 11$ .
$(2 * 5) * 1 = 2 * 1 = 4 - 1 + 3 = 6$ .
$5 * 1 = 10 - 1 + 3 = 12$ , so $2 * (5 * 1) = 2 * 12 = 4 - 12 + 3 = - 5$ .

Two observations: the order of $a$ and $b$ matters ( $2 * 5 \ne 5 * 2$ ), and so does the position of the brackets ( $6 \ne -5$ ). The first property is studied now; the second in the next lesson.

Definition

Commutative operation: a binary operation $*$ on $E$ is commutative if

a * b = b * a \quad \text{for all } a, b \in E .

Examples: addition and multiplication of real numbers, $\cup$ and $\cap$ are commutative. Subtraction and division are not: $7 - 2 \ne 2 - 7$ .

Worked example 3: proving and disproving commutativity

Which of these operations on $\mathbb{R}$ are commutative? (a) $a * b = a + b - 3 a b$ (b) $a \circ b = a + 2b$ (c) $a \star b = \max(a, b)$ , the larger of $a$ and $b$ .

Solution

For all real $a, b$ : $b * a = b + a - 3 b a = a + b - 3 a b = a * b$ , because real addition and multiplication are commutative. So $*$ is commutative.
A counterexample: $1 \circ 0 = 1$ but $0 \circ 1 = 2$ . So $\circ$ is not commutative.
The larger of two numbers does not depend on the order in which they are named, so $a \star b = b \star a$ : commutative.

Operation tables

On a small finite set, an operation can be listed in full in an operation table (also called a Cayley table). The entry in row $a$ and column $b$ is $a * b$ . Take $E = \{1, 3, 5, 7\}$ with $a \otimes b =$ the remainder when $a b$ is divided by 8. For instance $3 \otimes 5 = 15 - 8 = 7$ and $5 \otimes 7 = 35 - 4 \times 8 = 3$ :

$\otimes$	1	3	5	7
1	1	3	5	7
3	3	1	7	5
5	5	7	1	3
7	7	5	3	1

Two properties can be read straight from a table:

Closure: every entry belongs to $E$ . Here every entry is 1, 3, 5 or 7, so $E$ is closed under $\otimes$ .
Commutativity: the table is symmetric about its leading diagonal (top-left to bottom-right): the entry in row $a$ , column $b$ equals the entry in row $b$ , column $a$ . This table is symmetric, so $\otimes$ is commutative.

Tip

To test a table for commutativity, fold it in your mind along the leading diagonal: every entry must land on an equal entry. One mismatch is a counterexample.

Gopius E-Learning: Introduction to Binary Operation and Properties. Watch for the definition of a binary operation and how closure and commutativity are tested on given operations; associativity and distributivity come in the next lesson.

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Practice

Justify every answer: a proof with letters, or a counterexample.

Which of these are binary operations on the given set? (a) $\times$ on the odd integers (b) $+$ on the odd integers (c) $a - b$ on $\mathbb{Z}$ (d) $a \div b$ on $\mathbb{R}$ .
On $\mathbb{Z}$ , $a \circ b = 3a + b - 2$ . Find $4 \circ (-1)$ and $(-1) \circ 4$ . Is $\circ$ commutative?
Show that $a * b = a b + a + b$ is a commutative binary operation on $\mathbb{Z}$ .
On $\mathbb{Z}$ , $a * b = a + b + 4$ . Solve (a) $x * 3 = 10$ (b) $x * x = 0$ .
On $E = \{1, 2, 3, 4\}$ , let $a \odot b$ be the remainder when $a b$ is divided by 5. Draw the operation table. Is $E$ closed? Is $\odot$ commutative?
On $\mathbb{N}$ , $a \triangleright b = a$ (the left element always wins). Is it a binary operation? Is it commutative?
Explain why the average $a \diamond b = \frac{a + b}{2}$ is a binary operation on $\mathbb{Q}$ but not on $\mathbb{Z}$ .
For which real numbers $k$ is $a * b = a + b + k a b^{2}$ commutative on $\mathbb{R}$ ?

Show answersHide answers

(a) Yes: odd × odd is odd. (b) No: $1 + 3 = 4$ is even. (c) Yes. (d) No: $a \div 0$ is undefined.
$4 \circ (-1) = 12 - 1 - 2 = 9$ ; $(-1) \circ 4 = -3 + 4 - 2 = -1$ . Not commutative.
For integers $a, b$ , $a b + a + b$ is an integer (closure), and $b * a = b a + b + a = a b + a + b = a * b$ .
(a) $x + 7 = 10$ , $x = 3$ (b) $2 x + 4 = 0$ , $x = - 2$ .
Rows (for 1, 2, 3, 4): 1, 2, 3, 4 / 2, 4, 1, 3 / 3, 1, 4, 2 / 4, 3, 2, 1. Every entry is in $E$ (closed) and the table is symmetric (commutative).
Yes, $a \in \mathbb{N}$ always. Not commutative: $2 \triangleright 3 = 2$ but $3 \triangleright 2 = 3$ .
The average of two rationals is rational, so $\mathbb{Q}$ is closed; but $1 \diamond 2 = \frac{3}{2} \notin \mathbb{Z}$ .
We need $k a b^{2} = k b a^{2}$ , that is $k a b (b - a) = 0$ , for all $a, b$ . With $a = 1, b = 2$ this gives $2 k = 0$ , so $k = 0$ (and $k = 0$ works).

Quick check

Which of these is NOT a binary operation on ℕ = {0, 1, 2, …}?

Key points

A binary operation on $E$ combines any two elements of $E$ into exactly one element of $E$ (closure).
Commutative means $a * b = b * a$ for all $a, b$ ; in a table, symmetry about the leading diagonal.
One counterexample disproves a property; a proof with letters is needed to establish it.

Sources and credits

Written by Ganzaa: the explanations, the diagram, the worked examples, the practice items and the questions. The unit follows the topic list of REB's S4 Core Mathematics (2026–2027); no REB text is reproduced.
Video: Introduction to Binary Operation and Properties: Closure, Commutative, Associative, Distributive, Gopius E-Learning, embedded from YouTube.

1Arithmetic on a clock face

2The set matters

3Operation tables

Arithmetic on a clock face

The set matters

Operation tables