1.

1: Binary operations
Binary operation
Definition: Binary operation
Let S be a non-empty set, and ⋆ said to be a binary operation on S , if a ⋆ b is defined for all a, b ∈ S . In other words, ⋆ is a
rule for any two elements in the set S .

Example 1.1.1:
The following are binary operations on Z:
1. The arithmetic operations, addition +, subtraction −, multiplication ×, and division ÷.
2. Define an operation oplus on Z by a ⊕ b = ab + a + b, ∀a, b ∈ Z .
3. Define an operation ominus on Z by a ⊖ b = ab + a − b, ∀a, b ∈ Z .
4. Define an operation otimes on Z by a ⊗ b = (a + b)(a + b), ∀a, b ∈ Z .
5. Define an operation oslash on Z by a ⊘ b = (a + b)(a − b), ∀a, b ∈ Z .
6. Define an operation min on Z by a ∨ b = min{a, b}, ∀a, b ∈ Z .
7. Define an operation max on Z by a ∧ b = max{a, b}, ∀a, b ∈ Z .
8. Define an operation defect on Z by a ∗ b = a + b − 3, ∀a, b ∈ Z .
3

Lets explore the binary operations, before we proceed:

Example 1.1.2:
1. 2 ⊕ 3 = (2)(3) + 2 + 3 = 11 .
2. 2 ⊗ 3 = (2 + 3)(2 + 3) = 25 .
3. 2 ⊘ 3 = (2 + 3)(2 − 3) = −5 .
4. 2 ⊖ 3 = (2)(3) + 2 − 3 = 5 .
5. 2 ∨ 3 = 2 .
6. 2 ∧ 3 = 3 .

Exercise 1.1.2
1. −2 ⊕ 3 .
2. −2 ⊗ 3 .
3. −2 ⊘ 3 .
4. −2 ⊖ 3 .
5. −2 ∨ 3 .
6. −2 ∧ 3 .

Properties:

Closure property

Definition: Closure
Let S be a non-empty set. A binary operation ⋆ on S is said to be a closed binary operation on S , if a ⋆ b ∈ S, ∀a, b ∈ S .

Below we shall give some examples of closed binary operations, that will be further explored in class.

Example 1.1.3: Closed binary operations

The following are closed binary operations on Z.

1. The addition +, subtraction −, and multiplication ×.

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 2. Define an operation oplus on Z by a ⊕ b = ab + a + b, ∀a, b ∈ Z .
3. Define an operation ominus on Z by a ⊖ b = ab + a − b, ∀a, b ∈ Z .
4. Define an operation otimes on Z by a ⊗ b = (a + b)(a + b), ∀a, b ∈ Z .
5. Define an operation oslash on Z by a ⊘ b = (a + b)(a − b), ∀a, b ∈ Z .
6. Define an operation min on Z by a ∨ b = min{a, b}, ∀a, b ∈ Z .
7. Define an operation max on Z by a ∧ b = max{a, b}, ∀a, b ∈ Z .
8. Define an operation defect on Z by a ∗ b = a + b − 3, ∀a, b ∈ Z .
3

Exercise 1.1.1

Determine whether the operation ominus on Z is closed?

+

Example 1.1.4: Counter Example

Division (÷ ) is not a closed binary operations on Z.

2, 3 ∈ Z but 2

3
∉ Z .

Summary of arithmetic operations and corresponding sets:

+ × − ÷

Z+ closed closed not closed not closed

Z closed closed closed not closed

closed (only when 0 is
Q closed closed closed
not included)
closed (only when 0 is
R closed closed closed
not included)

Associative property
Definition: Associative
Let S be a subset of Z. A binary operation ⋆ on S is said to be associative , if (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c), ∀a, b, c ∈ S .

We shall assume the fact that the addition (+) and the multiplication (×) are associative on Z . (You don't need to prove them!).
+

Below is an example of proof when the statement is True.

Example 1.1.5: Associative

Determine whether the binary operation oplus is associative on Z.

We shall show that the binary operation oplus is associative on Z.

Below is an example of how to disprove when a statement is False.

Example 1.1.6: Not Associative

Determine whether the binary operation subtraction (−) is associative on Z.

Answer: The binary operation subtraction (−) is not associative on Z.

Commutative property

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 Definition: Commutative property
Let S be a non-empty set. A binary operation ⋆ on S is said to be commutative, if a ⋆ b = b ⋆ a, ∀a, b ∈ S .

We shall assume the fact that the addition (+) and the multiplication( × ) are commutative on Z+ . (You don't need to prove
them!).
Below is the proof of subtraction (−) NOT being commutative.

Example 1.1.7: NOT Commutative

Determine whether the binary operation subtraction − is commutative on Z.

Example 1.1.8: Commutative

Determine whether the binary operation oplus is commutative on Z.

We shall show that the binary operation oplus is commutative on Z.

Identity
Definition: Identity
A non-empty set S with binary operation ⋆, is said to have an identity e ∈ S , if e ⋆ a = a ⋆ e = a, ∀a ∈ S.

Note that 0 is called additive identity on (Z, +), and 1 is called multiplicative identity on (Z, ×).

Example 1.1.9: Is identity unique?

Let S be a non-empty set and let ⋆ be a binary operation on S . If e and e are two identities in (S, ⋆), then e
1 2 1 = e2 .
Proof:
Suppose that e and e are two identities in (S, ⋆).
1 2

Then e 1 = e1 ⋆ e2 = e2 .

Hence identity is unique. □

Example 1.1.10: Identity

Does (Z, ⊕) have an identity?

Example 1.1.11:

Does (Z, ⊗) have an identity?

Distributive Property
Definition: Distributive property
Let S be a non-empty set. Let ⋆ and ⋆ be two different binary operations on S .
1 2

Then ⋆ is said to be distributive over ⋆ on S if a ⋆

1 2 1 (b ⋆2 c) = (a ⋆1 b) ⋆2 (a ⋆1 c), ∀a, b, c, ∈ S .

Note that the multiplication distributes over the addition on Z. That is, 4(10 + 6) = (4)(10) + (4)(6) = 40 + 24 = 64 .
Further, we extend to (a + b)(c + d) = ac + ad + bc + bd (FOIL).
F-First
O-Outer
I-Inner

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L-Last
This property is very useful to find (26)(27) as shown below:

Example 1.1.12: Find (26)(27)

20 6

20 400 120

7 140 42

Hence (26)(27) = 400 + 120 + 140 + 42 = 702 .

Let's play a game!
PhET: Area Model Multiplication

Example 1.1.13:

Does multiplication distribute over subtraction?

Example 1.1.14:

Does division distribute over addition?

Example 1.1.15:

Does ⊗ distribute over ⊕ on Z ?

Summary
In this section, we have learned the following for a non-empty set S :
1. Binary operation,
2. Closure property,
3. Associative property,
4. Commutative property,
5. Distributive property, and
6. Identity.

This page titled 1.1: Binary operations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini
Thangarajah.

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