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Rivera, Monica T

A binary operation is a rule for combining two elements from a set to produce another element from the same set, typically denoted as a * b. Key properties of binary operations include closure, commutativity, associativity, identity element, and inverse element, with examples such as addition and multiplication on integers and real numbers. Each example illustrates how these properties apply within their respective sets.

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Aaron Baluyut
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16 views6 pages

Rivera, Monica T

A binary operation is a rule for combining two elements from a set to produce another element from the same set, typically denoted as a * b. Key properties of binary operations include closure, commutativity, associativity, identity element, and inverse element, with examples such as addition and multiplication on integers and real numbers. Each example illustrates how these properties apply within their respective sets.

Uploaded by

Aaron Baluyut
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Binary Operations and Their

Properties
An Introduction with Examples
What is a Binary Operation?
• A binary operation is a rule for combining two
elements (a and b) from a set to produce another
element from the same set.

• It is usually denoted as: a * b

Examples of binary operations:


• - Addition: a + b
• - Multiplication: a × b
• - Matrix multiplication (defined over matrices)
Properties of Binary Operations
• 1. **Closure**: If a and b are in set S, then a *
b is also in S.
• 2. **Commutativity**: a * b = b * a
• 3. **Associativity**: (a * b) * c = a * (b * c)
• 4. **Identity Element**: There exists e such
that a * e = a = e * a
• 5. **Inverse Element**: For every a, there
exists a' such that a * a' = e
Example 1: Addition on Integers (ℤ,
+)
• - Operation: a + b, where a, b ∈ ℤ
• - Closure: Always results in another integer
• - Commutative: a + b = b + a
• - Associative: (a + b) + c = a + (b + c)
• - Identity Element: 0 (a + 0 = a)
• - Inverse Element: For a, inverse is -a (a + (-a) =
0)
Example 2: Multiplication on Real
Numbers (ℝ, ×)
• - Operation: a × b, where a, b ∈ ℝ
• - Closure: Always results in another real
number
• - Commutative: a × b = b × a
• - Associative: (a × b) × c = a × (b × c)
• - Identity Element: 1 (a × 1 = a)
• - Inverse Element: For a ≠ 0, inverse is 1/a (a ×
1/a = 1)
Binary Operations and Their
Properties
An Introduction with Examples

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