Binary Operations in Discrete Mathematics: Combining Set Elements
Explore binary operations in discrete mathematics, focusing on how they combine two elements from a set to produce a single output within the same set. This guide explains the concept of binary operations, their representation using tables, and the generalization to n-ary operations.
Binary Operations in Discrete Mathematics
What is a Binary Operation?
A binary operation combines two elements from a set to produce a single element, which is also within that same set. It's a function that takes two inputs and gives one output. The operation is denoted by a symbol placed between the two elements (operands).
Examples of Binary Operations
- Addition (+) on natural numbers (ℕ): Adding any two natural numbers results in another natural number.
- Subtraction (-) on integers (ℤ): Subtracting any two integers results in an integer. (Note: Subtraction on natural numbers is not a binary operation because the result could be negative.)
- Multiplication (×) on natural numbers, integers, and real numbers (ℝ): Multiplying any two numbers from these sets gives another number in the same set.
- Union (∪) on sets: The union of any two sets is a set.
- Intersection (∩) on sets: The intersection of any two sets is a set.
Representing Binary Operations with Tables
For a finite set A = {a₁, a₂, ..., an}, a binary operation * can be represented using a table. The rows and columns represent the elements of A, and the entry at row ai and column aj shows the result of ai * aj.
Example: Binary Operation Table
Consider the set A = {1, 2, 3} with the binary operation * defined as a * b = 2a + b.
| * | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 3 | 4 | 5 |
| 2 | 5 | 6 | 7 |
| 3 | 7 | 8 | 9 |
N-ary Operations
An n-ary operation is a generalization of a binary operation. It takes n inputs from a set A and produces a single output, also in A. A binary operation is simply a 2-ary operation.
Conclusion
Binary operations are fundamental to algebra. Understanding how they're defined and represented, along with their properties, is crucial for studying various algebraic structures.