This document discusses Boolean algebra and logic gates. It begins with basic definitions of Boolean algebra, including elements, operators, axioms, and binary operations. It then covers axiomatic definitions of Boolean algebra, two-valued Boolean algebra using 0 and 1, basic theorems and properties including duality and DeMorgan’s theorems.

Monotone laws

It follows from the first five pairs of axioms that any complement is unique. The relation ≤ defined by a ≤ b if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤. It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that Additional reading material for laws and theorems of Boolean algebra. Chapter 1 presents the various binary systems suitable for representing information in digital systems.

Boole’s formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington.

Digital logic

When used in expressions, the operators are applied according to the precedence rules. As with elementary algebra, expressions in parentheses are evaluated first, following the precedence rules. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way.

The Minimal Set of Axioms

• When used in expressions, the operators are applied according to the precedence rules.
• Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington.
• Boolean algebra is the category of algebra in which the variable’s values are the truth values, true and false, ordinarily denoted 1 and 0 respectively.
• In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data.
• Chapter 2 introduces the basic postulates of Boolean algebra and shows the correlation between Boolean expressions and their corresponding logic diagrams.

Boolean terms such as x ∨ y become propositional formulas P ∨ Q; 0 becomes false or ⊥, and 1 becomes true or ⊤. It is convenient when referring to generic propositions to use Greek letters Φ, Ψ, … As metavariables (variables outside the language of propositional calculus, used when talking about propositional calculus) to denote propositions. Moreover, the number of equations needed can be further reduced. To begin with, some of the above laws are implied by some of the others. A sufficient subset of the above laws consists of the pairs of associativity, commutativity, and absorption laws, distributivity of ∧ over ∨ (or the other distributivity law—one suffices), and the two complement laws.

Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff’s 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models. These are simply syntactic sugar and can all be expressed in terms of conjunction, disjunction, and negation. Since these laws are so fundamental, they are used very often (especially material implication). These truth tables can be derived from the axioms and some laws we prove in the next post. The negation truth table is a consequence of identity and annihilation laws.

8 Timing Diagram

Boolean algebra is not sufficient to capture logic formulas using quantifiers, like those from first-order logic. A precursor of Boolean algebra was Gottfried Wilhelm Leibniz’s algebra of concepts. The usage of binary in relation to the I Ching was central to Leibniz’s characteristica universalis. It eventually created the foundations of algebra of concepts. Leibniz’s algebra of concepts is deductively equivalent to the Boolean algebra of sets.

• Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages.
• The laws complementation 1 and 2, together with the monotone laws, suffice for this purpose and can therefore be taken as one possible complete set of laws or axiomatization of Boolean algebra.
• Boolean Algebra is a branch of mathematics that deals with variables that have only two possible values — typically denoted as 0 and 1 (or false and true).
• Boolean Algebra is closely related to Set Theory, as every power set of a set forms a Boolean algebra under union, intersection, and complementation.
• The Boolean algebras so far have all been concrete, consisting of bit vectors or equivalently of subsets of some set.

2 Bases and Number Systems

(In older works, some authors required 0 and 1 to be distinct elements in order to exclude this case.)citation needed 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). This chapter outlines the formal procedures for the analysis and design of combinational circuits. Some basic components used in the design of digital systems, such as adders and code converters, are introduced as design examples. Frequently used digital logic functions such as parallel adders and subtractors, decoders, encoders, and multiplexers are explained, and their use in the design of combinational circuits is illustrated. HDL examples are given in gate‐level, dataflow, and behavioral models to show the alternative ways available for describing combinational circuits in Verilog HDL.

The Associative Law states that when three or more variables are combined using the AND or OR operators, the grouping of the variables does not affect the result. This law allows us to regroup terms without changing the axiomatic definition of boolean algebra output. The Commutative Law states that the order in which two variables are combined using the AND or OR operators does not affect the result.

Combinational Circuits

Let $\neg$ be a unary operation (takes one variable as an argument). Let $\vee$ and $\wedge$ be binary operations (take two variables as arguments). Let $a$ be a boolean variable and $A, B$ be boolean expressions. In the Boolean Algebra, we have identity elements for both AND(.) and OR(+) operations. The identity law states that in Boolean algebra, we have such variables that, on operating with the AND and OR operations we get the same result, i.e. Boolean Algebra is a branch of mathematics that deals with variables that have only two possible values — typically denoted as 0 and 1 (or false and true).

In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, hence in both algebras it satisfies the double negation law (also called involution law) The complement operation is defined by the following two laws. These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.

The Boolean algebras so far have all been concrete, consisting of bit vectors or equivalently of subsets of some set. Such a Boolean algebra consists of a set and operations on that set which can be shown to satisfy the laws of Boolean algebra. But if in addition to interchanging the names of the values, the names of the two binary operations are also interchanged, now there is no trace of what was done. The end product is completely indistinguishable from what was started with. The columns for x ∧ y and x ∨ y in the truth tables have changed places, but that switch is immaterial.

4 NMOS and PMOS Logic Gates

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. The lines on the left of each gate represent input wires or ports.