# Axioms and Laws of Boolean Algebra

Even the theory of Boolean algebras with a

distinguished ideal is decidable. On the other hand, the theory of a

Boolean algebra with a distinguished subalgebra is undecidable. Both

the decidability results and undecidablity results extend in various

ways to Boolean algebras in extensions of first-order logic. The sets of logical expressions are known as Axioms or postulates of Boolean Algebra.

- This opened a new and

fruitful perspective, deviating from the traditional approach to

logic, where for centuries scholars had struggled to come up with

clever mnemonics to memorize a very small catalog of valid conversions

and syllogisms and their various interrelations. - In this way when talking about different equational theories we can push the rules to one side as being independent of the particular theories, and confine attention to the axioms as the only part of the axiom system characterizing the particular equational theory at hand.
- Become propositional variables (or atoms) P,Q,…, Boolean terms such as x∨y become propositional formulas P∨Q, 0 becomes false or ⊥, and 1 becomes true or T.
- If we define a homologue of an algebra to be a model of the equational theory of that algebra, then a Boolean algebra can be defined as any homologue of the prototype.

However, he refined his

system of axioms and rules of inference until the result was

essentially the modern system of Boolean algebra for ground

terms, that is, terms where the class symbols are to be thought

of as constants, not as variables. Many axiomatic systems were developed in the nineteenth century, including non-Euclidean geometry, the foundations of real analysis, Cantor’s set theory, axiomatic definition of boolean algebra Frege’s work on foundations, and Hilbert’s ‘new’ use of axiomatic method as a research tool. For example, group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies.

## De-Morgan’s Law

Zermelo–Fraenkel set theory, with the historically controversial axiom of choice included, is commonly abbreviated ZFC, where “C” stands for “choice”. Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.[5] Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. The system has at least two different models – one is the natural numbers (isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with the cardinality of the continuum). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, the property distinguishing these models is their cardinality — a property which cannot be defined within the system. A model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system.

This makes the integers a Boolean algebra, with union being bit-wise OR and complement being −x−1. There are only countably many integers, so this infinite Boolean algebra is countable. Another way of describing this algebra is as the set of all finite and cofinite sets of natural numbers, with the ultimately all-ones sequences corresponding to the cofinite sets, those sets omitting only finitely many natural numbers. The algebra of logic, as an explicit algebraic

system showing the underlying mathematical structure of logic, was

introduced by George Boole (1815–1864) in his book The

Mathematical Analysis of Logic (1847). It is therefore to be

distinguished from the more general approach of algebraic

logic. The methodology initiated by Boole was successfully

continued in the 19th century in the work of William

Stanley Jevons (1835–1882), Charles Sanders Peirce

(1839–1914), Ernst Schröder (1841–1902), among many

others, thereby establishing a tradition in (mathematical) logic.

## 1854—Boole’s Final Presentation of his Algebra of Logic

Mathematicians decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated. Beyond consistency, relative consistency is also the mark of a worthwhile axiom system. This describes the scenario where the undefined terms of a first axiom system are provided definitions from a second, such that the axioms of the first are theorems of the second. In an axiomatic system, an axiom is called independent if it cannot be proven or disproven from other axioms in the system. A system is called independent if each of its underlying axioms is independent.

- The operations include the variables, for example 1f2 is x0 while 2f10 is x0 (as two copies of its unary counterpart) and 2f12 is x1 (with no unary counterpart).
- The customary formulation of an axiom system consists of a set of axioms that “prime the pump” with some initial identities, along with a set of inference rules for inferring the remaining identities from the axioms and previously proved identities.
- Further work has been done for reducing the number of axioms; see Minimal axioms for Boolean algebra.
- Since the

application of symbolical algebra to differential equations had

proceeded through the introduction of differential operators, it must

have been natural for Boole to look for operators that applied in the

area of Aristotelian logic. - Volume II augments the algebra of logic for classes developed in

Volume I so that it can handle existential statements. - The Boolean algebras we have seen so far have all been concrete, consisting of bit vectors or equivalently of subsets of some set.

For his single rule of inference Jevons chose his principle of

substitution—in modern terms this was essentially a combination

of ground replacement and transitivity. He showed how to derive

transitivity of equality from this; he could have derived symmetry as

well but did not. The associative law was missing—it was

implicit in the lack of parentheses in his expressions. Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing the axiomatic method.

## Peirce: Basing the Algebra of Logic on Subsumption

In contrast, in a list of some but not all of the same laws, there could have been Boolean laws that did not follow from those on the list, and moreover there would have been models of the listed laws that were not Boolean algebras. In the other direction, there may exist many homomorphisms from a Boolean algebra B to 2. Any such homomorphism partitions B into those elements mapped to 1 and those to 0. When B is finite its ultrafilters pair up with its atoms; one atom is mapped to 1 and the rest to 0. Each ultrafilter of B thus consists of an atom of B and all the elements above it; hence exactly half the elements of B are in the ultrafilter, and there as many ultrafilters as atoms.

Volume II augments the algebra of logic for classes developed in

Volume I so that it can handle existential statements. First, using

modern semantics, Schröder proved that one cannot use equations

to express “Some \(X\) is \(Y\)”. However, he noted that

one can easily express it with a negated equation, namely \(XY \ne

0\). Volume II, a study of the calculus of classes using both

equations and negated equations, attempted to cover the same topics

covered in Vol. I, in particular there was considerable effort devoted

to finding an Elimination Theorem.

## Axioms and Laws of Boolean Algebra

Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem. In this article, we are going to discuss Axioms of Boolean Algebra; these axioms/Theorems are important as these will be used in many different topics of Digital Electronics like Sequential Circuit Designing and Combinational Circuit Designing as well. The triangle denotes the operation that simply copies the input to the output; the small circle on the output denotes the actual inversion complementing the input. The convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through, whether it is an input or output port.

Later commentators would gloss over this chapter, and no one

seems to have worked through its details. In late 1847, Boole and Augustus De Morgan (1806–1871) each

published a book on logic—Boole’s Mathematical Analysis of

Logic (1847) and De Morgan’s Formal Logic (1847). De

Morgan’s approach was to dissect every aspect of traditional deductive

logic (usually called ‘Aristotelian logic’) into its

minutest components, to consider ways to generalize these components,

and then, in some cases, undertake to build a logical system using

these components. Unfortunately, he was never able to incorporate his

best ideas into a significant system. His omission of a symbol for

equality made it impossible to develop an equational algebra of logic. In mathematics, axiomatization is the process of taking a body of knowledge and working backwards towards its axioms.

For so-called “active-high” logic, 0 is represented by a voltage close to zero or “ground”, while 1 is represented by a voltage close to the supply voltage; active-low reverses this. The line on the right of each gate represents the output port, which normally follows the same voltage conventions as the input ports. Then it would still be Boolean algebra, and moreover operating on the same values. However it would not be identical to our original Boolean algebra because now we find ∨ behaving the way ∧ used to do and vice versa. So there are still some cosmetic differences to show that we’ve been fiddling with the notation, despite the fact that we’re still using 0s and 1s.

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A basis is any set from which the remaining operations can be obtained by composition. Three bases for Boolean algebra are in common use, the lattice basis, the ring basis, and the Sheffer stroke or NAND basis. These bases impart respectively a logical, an arithmetical, and a parsimonious character to the subject. A model of a theory is an algebra interpreting the operation symbols in the language of the theory and satisfying the equations of the theory.