Axioms and Laws of Boolean Algebra

Propositional calculus restricts attention to abstract propositions, those built up from propositional variables using Boolean operations. Instantiation is still possible within propositional calculus, but only by instantiating propositional variables by abstract propositions, such as instantiating Q by Q → P in P → (Q → P) to yield the instance P → ((Q → P) → P). The term “algebra” denotes both a subject, namely the subject of algebra, and an object, namely an algebraic structure. Whereas the foregoing has addressed the subject of Boolean algebra, this section deals with mathematical objects called Boolean algebras, defined in full generality as any model of the Boolean laws.

Values

All concrete Boolean algebras satisfy the laws (by proof rather than fiat), whence every concrete Boolean algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set and certain operations satisfying certain laws or axioms by fiat is entirely analogous to the abstract definitions of group, ring, field etc. characteristic of modern or abstract algebra. The sets of logical expressions are known as Axioms or postulates of Boolean Algebra. An axiom is nothing more than the definition of three basic logic operations (AND, OR, and NOT).

The other regions are left unshaded to indicate that x ∧ y is 0 for the other three combinations. In order to define a Boolean algebra, we need the additional concept of complementation. A lattice axiomatic definition of boolean algebra must have both a greatest element and a least element in order for complementation to take place. The following definition will save us some words in the rest of this section.

Relations and Functions

It follows from the first five pairs of axioms that any complement is unique. Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see Proven properties). Boolean Algebra also called Logical Algebra is a branch of mathematics that deals with Boolean Variables such as, 0 and 1. Boolean Algebra is vital in AI, notably in the construction of decision-making algorithms and neural networks.

Basics of Algebra

1. The shading indicates the value of the operation for each combination of regions, with dark denoting 1 and light 0 (some authors use the opposite convention).
2. It is formed by combining Boolean variables and logical expressions such as AND, OR, and NOT.
3. It’s used to model logical thinking and decision trees, which are crucial in machine learning and expert systems.
4. A minimum number of gates in a Boolean expression reduces the complexity of a circuit.
5. Boolean algebra is used to develop complex search queries in legal databases.
6. Rather than attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages per wire, high and low.

As metavariables (variables outside the language of propositional calculus, used when talking about propositional calculus) to denote propositions. Therefore it can be inferred that Boolean Algebra in its axioms and theorems acts as the basis on which digital electronics mainly builds sequential and combinational circuits. If these axioms as; Commutative, Associative, Distributive, Idempotence, and Absorption are learned, complicated Boolean expressions can be simplified and this results in efficient circuit designs. This is opposed to arithmetic algebra where a result may come out to be some number different from 0 or 1 showing the binary nature of Boolean operations and confirming that Boolean logic is distinctive in digital systems. Boolean algebra can be used very efficiently in complex decision-making processes by decomposing complex expressions into simple logic(resultant output will be same).

While we have not shown the Venn diagrams for the constants 0 and 1, they are trivial, being respectively a white box and a dark box, neither one containing a circle. However, we could put a circle for x in those boxes, in which case each would denote a function of one argument, x, which returns the same value independently of x, called a constant function. When values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously, the members of each pair are called dual to each other. The duality principle, also called De Morgan duality, asserts that Boolean algebra is unchanged when all dual pairs are interchanged. The “pairings” of the boolean algebra laws reminds us of the principle of duality, which we state for a Boolean algebra.

A variable or the complement of the variable in Boolean Algebra is called the Literal. Literals are the basic building blocks of the boolean expressions and functions. Syntactically, every Boolean term corresponds to a propositional formula of propositional logic. In this translation between Boolean algebra and propositional logic, Boolean variables x, y, … Boolean terms such as x ∨ y become propositional formulas P ∨ Q; 0 becomes false or ⊥, and 1 becomes true or T. It is convenient when referring to generic propositions to use Greek letters Φ, Ψ, …