Mathematics - Discrete Mathematics - Binary Operations

Introduction

Hey it's a me again @drifter1!

Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Binary Operations.

So, without further ado, let's get straight into it!

Binary Operations

For a non-empty set A a binary operation is a function of the form f : A x A → A. The two domains and the co-domain are the same exact set A. So, basically a binary operation is any rule that combines two elements (operands) in order to produce another element. For example, the arithmetic operations of addition, subtraction and multiplication are all binary operations on the set of integers.

The same logic can also be extended towards vector addition and matrix multiplication, by considering so called n-ary operations of the form f : A x A x ... x A → A.

Any binary operation can also be represented by a table of all possible combinations and their corresponding results. Considering the set of natural numbers, the binary operation of addition "+" can be represented by the table:

Properties

Binary operations may also satisfy lots of properties...

Closure

For a non-empty set A and binary operation "*" on A, the binary operation satisfies closure when the result of the binary operation a * b ∈ A. Of course a, b ∈ A. For example, addition on the set of integers is a closed operation.

Associativity

For a non-empty set A and binary operation "*" on A, the binary operation satisfies associativity if for every a, b, c ∈ A:

Commutativity

For a non-empty set A and binary operation "*" on A, the binary operation satisfies commutativity if for every a, b ∈ A, a * b = b * a.

Distributivity

For a non-empty set A and binary operation "*" on A, the binary operation satisfies distributivity if for every a, b, c ∈ A:

which are the left and right distributivity respectively.

Identity

For a non-empty set A and binary operation "*" on A, the binary operation satisfies the identity property if for every a ∈ A there exists an element e ∈ A, so that:

which are known as the right and left identities respectively.

Inverse

For a non-empty set A and binary operation "*" on A, the binary operation satisfies the inverse property if for every a ∈ A there exists an element b ∈ A, so that:

which are known as the right and left inverse respectively.

Idempotent

For a non-empty set A and binary operation "*" on A, the binary operation satisfies the idempotent property if for every a ∈ A : a * a = a.

Cancellation

For a non-empty set A and binary operation "*" on A, the binary operation satisfies the cancellation property if for every a, b, c ∈ A:

which are the left and right cancellation respectively.

RESOURCES:

References

Images

Mathematical equations used in this article, have been generated using quicklatex.

Block diagrams and other visualizations were made using draw.io.

Previous articles of the series

Introduction → Discrete Mathematics, Why Discrete Math, Series Outline
Sets → Set Theory, Sets (Representation, Common Notations, Cardinality, Types)
Set Operations → Venn Diagrams, Set Operations, Properties and Laws
Sets and Relations → Cartesian Product of Sets, Relation and Function Terminology (Domain, Co-Domain and Range, Types and Properties)
Relation Closures → Relation Closures (Reflexive, Symmetric, Transitive), Full-On Example
Equivalence Relations → Equivalence Relations (Properties, Equivalent Elements, Equivalence Classes, Partitions)
Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
Combinatorial Principles → Combinatorics, Basic Counting Principles (Additive, Multiplicative), Inclusion-Exclusion Principle (PIE)
Combinations and Permutations → Factorial, Binomial Coefficient, Combination and Permutation (with / out repetition)
Combinatorics Topics → Pigeonhole Principle, Pascal's Triangle and Binomial Theorem, Counting Derangements
Propositions and Connectives → Propositional Logic, Propositions, Connectives (∧, ∨, →, ↔ and ¬)
Implication and Equivalence Statements → Truth Tables, Implication, Equivalence, Propositional Algebra
Proof Strategies (part 1) → Proofs, Direct Proof, Proof by Contrapositive, Proof by Contradiction
Proof Strategies (part 2) → Proof by Cases, Proof by Counter-Example, Mathematical Induction
Sequences and Recurrence Relations → Sequences (Terms, Definition, Arithmetic, Geometric), Recurrence Relations
Probability → Probability Theory, Probability, Theorems, Example
Conditional Probability → Conditional Probability, Law of Total Probability, Bayes' Theorem, Full-On Example
Graphs → Graph Theory, Graphs (Vertices, Types, Handshake Lemma)
Graphs 2 → Graph Representation (Adjacency Matrix and Lists), Graph Types and Properties (Isomorphic, Subgraphs, Bipartite, Regular, Planar)
Paths and Circuits → Paths, Circuits, Euler, Hamilton
Trees → Trees (Rooted, General and Binary), Tree Traversal, Spanning Trees
Common Graph Problems → Shortest Path Problem, Graph Connectivity, Travelling Salesman Problem, Minimum Spanning Tree, Maximum Network Flow, Graph Coloring