[[Image 1](https://commons.wikimedia.org/wiki/File:Truth_table_for_AND,_OR,_and_NOT.png)]
## Introduction
Hey it's a me again [@drifter1](https://peakd.com/@drifter1)!
Today we continue with **Mathematics**, and more specifically the branch of "**Discrete Mathematics**", in order to get into **Boolean Algebra**.
This will be a small introduction to the topic. For more information I highly suggest checking out Logic Design in general.
So, without further ado, let's get straight into it!
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## Boolean Algebra
Boolean algebra is a branch of mathematics, which studies the algebra of 0 and 1, or false and true. It finds many applications in computer science, with logical circuits being the most direct application, but it's also useful in programming / coding in general.
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## Boolean Operations
The three main boolean operations are:
- AND (conjunction)
- OR (disjunction)
- NOT (negation)
These are the same exact operations that were covered in the article about Propositional Logic.
### Notation
In Boolean algebra the AND and OR operations are commonly symbolized using multiplication and addition respectively, meaning that AND uses "*" or even nothing, and OR uses "+". Using the symbols ∧ and ∨ is of course also valid. Lastly, negation is usually represented by an apostrophe (') or an overline (-) instead of an tilde (~) in front of the respective boolean variable.
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## Boolean Properties
Boolean operations satisfy the properties shown in the table below.
Boolean algebra is basically a complemented, distributive lattice, with boolean operations AND (∧ or *), OR (∨ or +) and a unique complement or unary operation defined for each element. The elements are only two: 0 and 1.
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## Boolean Expressions
Boolean expressions are expressions defined over boolean Algebra. Each element in such an expression is an boolean expression itself (i.e. 0 and 1 are boolean expressions on their own).
For example:
is a valid boolean expressions with boolean variables x and y.
Such expressions are evaluated by assigning either 0 or 1 to each boolean variable, yielding a final result of 0 or 1.
### Equivalent Expressions
Of course, there can be multiple boolean expressions giving the same result for the same assignment of values. These so called equivalent expressions have the same truth table.
### Canonical Forms
It's common to write boolean expressions in either disjunctive or conjunctive normal form, and so basically as a sum of products (SOP) or a product of sums (POS). In a SOP, the product terms are known as min-terms (resulting in 1), whilst in a POS the sum terms are called max-terms (resulting in 0).
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## RESOURCES:
### References
1. https://www.javatpoint.com/discrete-mathematics-tutorial
2. https://brilliant.org/wiki/discrete-mathematics/
### Images
1. [https://commons.wikimedia.org/wiki/File:Truth_table_for_AND,_OR,_and_NOT.png](https://commons.wikimedia.org/wiki/File:Truth_table_for_AND,_OR,_and_NOT.png)
Mathematical equations used in this article, have been generated using [quicklatex](http://quicklatex.com/).
Block diagrams and other visualizations were made using [draw.io](https://app.diagrams.net/).
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## Previous articles of the series
* [Introduction](https://peakd.com/hive-163521/@drifter1/mathematics-an-introduction-to-discrete-mathematics) → Discrete Mathematics, Why Discrete Math, Series Outline
* [Sets](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-sets) → Set Theory, Sets (Representation, Common Notations, Cardinality, Types)
* [Set Operations](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-set-operations) → Venn Diagrams, Set Operations, Properties and Laws
* [Sets and Relations](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-sets-and-relations) → Cartesian Product of Sets, Relation and Function Terminology (Domain, Co-Domain and Range, Types and Properties)
* [Relation Closures](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-relation-closures) → Relation Closures (Reflexive, Symmetric, Transitive), Full-On Example
* [Equivalence Relations](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-equivalence-relations) → Equivalence Relations (Properties, Equivalent Elements, Equivalence Classes, Partitions)
* [Partial Order Relations and Sets](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-partial-order-relations-and-sets) → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
* [Combinatorial Principles](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-combinatorial-principles) → Combinatorics, Basic Counting Principles (Additive, Multiplicative), Inclusion-Exclusion Principle (PIE)
* [Combinations and Permutations](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-combinations-and-permutations) → Factorial, Binomial Coefficient, Combination and Permutation (with / out repetition)
* [Combinatorics Topics](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-combinatorics-topics) → Pigeonhole Principle, Pascal's Triangle and Binomial Theorem, Counting Derangements
* [Propositions and Connectives](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-propositions-and-connectives) → Propositional Logic, Propositions, Connectives (∧, ∨, →, ↔ and ¬)
* [Implication and Equivalence Statements](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-implication-and-equivalence-statements) → Truth Tables, Implication, Equivalence, Propositional Algebra
* [Proof Strategies (part 1)](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-proof-strategies-part-1) → Proofs, Direct Proof, Proof by Contrapositive, Proof by Contradiction
* [Proof Strategies (part 2)](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-proof-strategies-part-2) → Proof by Cases, Proof by Counter-Example, Mathematical Induction
* [Sequences and Recurrence Relations](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-sequences-and-recurrence-relations) → Sequences (Terms, Definition, Arithmetic, Geometric), Recurrence Relations
* [Probability](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-probability) → Probability Theory, Probability, Theorems, Example
* [Conditional Probability](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-conditional-probability) → Conditional Probability, Law of Total Probability, Bayes' Theorem, Full-On Example
* [Graphs](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-graphs) → Graph Theory, Graphs (Vertices, Types, Handshake Lemma)
* [Graphs 2](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-graphs-2) → Graph Representation (Adjacency Matrix and Lists), Graph Types and Properties (Isomorphic, Subgraphs, Bipartite, Regular, Planar)
* [Paths and Circuits](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-paths-and-circuits) → Paths, Circuits, Euler, Hamilton
* [Trees](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-trees) → Trees (Rooted, General and Binary), Tree Traversal, Spanning Trees
* [Common Graph Problems](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-common-graph-problems) → Shortest Path Problem, Graph Connectivity, Travelling Salesman Problem, Minimum Spanning Tree, Maximum Network Flow, Graph Coloring
* [Binary Operations](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-binary-operations) → Binary Operations (n-ary, Table Representation), Properties
* [Groups](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-groups) → Groups (Properties, Theorems, Finite and Infinite, Abelian, Cyclic, Product, Homo-, Iso- and Auto-morphism)
* [Group-like Structures (part 1)](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-group-like-structures-part-1) → Subgroups, Semigroups, Monoids
* [Group-like Structures (part 2)](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-group-like-structures-part-2) → Magma, Quasigroup, Groupoid
* [Rings](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-rings) → Rings (Axioms, Commutative and Non-Commutative, Semirings, Subrings, Rng)
* [Fields](https://peakd.com/hive-163521/@drifter1/mathematics-discrete-mathematics-fields) → Fields (Axioms, Subfields, Finite Fields, Field Extension)
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## Final words | Next up
And this is actually it for today's post!
Not sure about the next article's topic yet.
See ya!
Keep on drifting!
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