Ockham algebras | Lattice theory | Algebraic logic | Algebra
In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that: * (A, ∨, ∧, 0, 1) is a bounded distributive lattice, and * ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an involution that additionally satisfies De Morgan's laws) In a De Morgan algebra, the laws * ¬x ∨ x = 1 (law of the excluded middle), and * ¬x ∧ x = 0 (law of noncontradiction) do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a Boolean algebra. Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0). Thus ¬ is a dual automorphism of (A, ∨, ∧, 0, 1). If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as an involutive anti-automorphism, that is, a structure A = (A, ≤, ¬) such that: * (A, ≤) is a bounded distributive lattice, and * ¬¬x = x, and * x ≤ y → ¬y ≤ ¬x. De Morgan algebras were introduced by Grigore Moisil around 1935, although without the restriction of having a 0 and a 1. They were then variously called quasi-boolean algebras in the Polish school, e.g. by Rasiowa and also distributive i-lattices by . (i-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of Antonio Monteiro. De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic. The standard fuzzy algebra F = ([0, 1], max(x, y), min(x, y), 0, 1, 1 − x) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold. Another example is Dunn's 4-valued logic, in which false < neither-true-nor-false < true and false < both-true-and-false < true, while neither-true-nor-false and both-true-and-false are not comparable. (Wikipedia).
Linear Algebra: Continuing with function properties of linear transformations, we recall the definition of an onto function and give a rule for onto linear transformations.
From playlist MathDoctorBob: Linear Algebra I: From Linear Equations to Eigenspaces | CosmoLearning.org Mathematics
Group Definition (expanded) - Abstract Algebra
The group is the most fundamental object you will study in abstract algebra. Groups generalize a wide variety of mathematical sets: the integers, symmetries of shapes, modular arithmetic, NxM matrices, and much more. After learning about groups in detail, you will then be ready to contin
From playlist Abstract Algebra
Algebra for Beginners | Basics of Algebra
#Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. Table of Conten
From playlist Linear Algebra
Linear Algebra Proofs 15a: Eigenvectors with the Same Eigenvalue Form a Linear Space
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Proofs
Linear Algebra Vignette 2a: RREF - What It's For
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes
Boolean Algebra 3 – De Morgan’s Theorem
This video follows on from the one about simplifying complex Boolean expressions using the laws of Boolean algebra. In particular this video covers De Morgan’s theorem and how it can be applied, along with the other laws, to simplify complex Boolean expressions. It includes worked exampl
From playlist Boolean Algebra
Formal Logic by Augustus De Morgan (1847)
In this video I will show you my copy of Formal Logic by Augustus De Morgan. This book was published in 1847 and is considered a collectors item. Note my copy is very hard to read because of the condition of the book but I tried my best to show some of the pages of this old book. You can f
From playlist Book Reviews
EEVacademy #2 - Digital Logic Boolean & Demorgan's Theorems
Boolean Algebra & Demorgan's Theorems explained and how they are useful for circuit simplification. EEVblog Main Web Site: http://www.eevblog.com The 2nd EEVblog Channel: http://www.youtube.com/EEVblog2 Support the EEVblog through Patreon! http://www.patreon.com/eevblog EEVblog Amazon
From playlist EEVacademy
Linear Algebra Vignette 1b: The Dilation Operator (Has Important Applications)
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes
Verify De Morgan's Law using a Truth Table ~(p V q) = ~p ^ ~q
Verify De Morgan's Law using a Truth Table ~(p V q) = ~p ^ ~q If you enjoyed this video please consider liking, sharing, and subscribing. Udemy Courses Via My Website: https://mathsorcerer.com My FaceBook Page: https://www.facebook.com/themathsorcerer There are several ways that you ca
From playlist Logical Form and Logical Equivalence
Verify De Morgan's Law by Using a Truth Table: ~(p ^ q) = ~p V ~q
Verify De Morgan's Law by Using a Truth Table: ~(p ^ q) = ~p V ~q If you enjoyed this video please consider liking, sharing, and subscribing. Udemy Courses Via My Website: https://mathsorcerer.com My FaceBook Page: https://www.facebook.com/themathsorcerer There are several ways that yo
From playlist Logical Form and Logical Equivalence
Linear Algebra Vignette 3d: Easy Eigenvalues - Linearly Dependent Columns
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes
Boolean Algebra 1 – The Laws of Boolean Algebra
This computer science video is about the laws of Boolean algebra. It briefly considers why these laws are needed, that is to simplify complex Boolean expressions, and then demonstrates how the laws can be derived by examining simple logic circuits and their truth tables. It also shows ho
From playlist Boolean Algebra
How to Verify the Logical Equivalence using the Laws of Logic: ~(~p ^ q) ^ (p V q) = p
How to Verify the Logical Equivalence using the Laws of Logic: ~(~p ^ q) ^ (p V q) = p If you enjoyed this video please consider liking, sharing, and subscribing. Udemy Courses Via My Website: https://mathsorcerer.com My FaceBook Page: https://www.facebook.com/themathsorcerer There a
From playlist Logical Form and Logical Equivalence
What is Abstract Algebra? (Modern Algebra)
Abstract Algebra is very different than the algebra most people study in high school. This math subject focuses on abstract structures with names like groups, rings, fields and modules. These structures have applications in many areas of mathematics, and are being used more and more in t
From playlist Abstract Algebra
This Math Book Really Freaked Me Out
This book is free. I think it's in the public domain. See the link below. It is called Vector Algebra and Trigonometry and it was written by Hayward. Here is the book for free I think: https://books.google.com/books?id=WPMGAAAAYAAJ& Here is a reprint https://amzn.to/3BKxzyX Useful Math Su
From playlist Book Reviews
ELEC2141 Digital Circuit Design - Lecture 5
ELEC2141 Week 2 Lecture 2: Combinational Logic Circuits 1
From playlist ELEC2141 Digital Circuit Design
Linear Algebra Vignette 3a: Easy Eigenvalues - Introduction
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes
Linear Algebra Vignette 3e: Easy Eigenvalues - Triangular Matrices
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes
Introduction to Logically Equivalent Statements
This video introduces logically equivalent statements and defines De Morgan's laws, implications are disjunctions, double negation, and negation of an implication. mathispower4u.com
From playlist Symbolic Logic and Proofs (Discrete Math)