The three Venn diagrams in the figure below represent respectively conjunction x ∧ y, disjunction x ∨ y, and complement ¬x. To clarify, writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms, nor can it rule out any model of them. 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. The complement operation is defined by the following two laws.

  • A truth table represents all the combinations of input values and outputs in a tabular manner.
  • First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers.
  • So this example while not technically concrete is at least «morally» concrete via this representation, called an isomorphism.
  • Boole’s formulation differs from that described above in some important respects.
  • More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions.

Recall the definition of sup and inf from the section above on the
underlying partial order of a Boolean algebra. A complete
Boolean algebra is one every subset of which has both a sup and an inf. There exists a unique homomorphism from 2 to every Boolean
algebra, since any homomorphism must carry bottom to bottem and
top to top.


The three main logical operations of boolean algebra are conjunction, disjunction, and negation. A complete Boolean algebra is called completely
distributive when arbitrary conjunctions distribute over arbitrary
disjunctions and vice versa. A complete and completely distributive
Boolean algebra is isomorphic to a CABA, whence there do exist free
complete and completely distributive Boolean algebras. This neatly
rescues infinitary Boolean logic from the fate the Gaifman-Hales
result seemed to consign it to. A cofinite set of
integers is one that omits only finitely many integers. The finite
sets are clearly closed under finite union and finite intersection, and
likewise the cofinite sets are so closed.

The two halves of a sequent are called the antecedent and the succedent respectively. The antecedent is interpreted as the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of the succedent by the antecedent. The semantics of propositional logic rely on truth assignments. In classical semantics, only the two-element Boolean algebra is used, while in Boolean-valued semantics arbitrary Boolean algebras are considered.

  • The inverse of the boolean variable is called the complement of the variable.
  • Boolean algebra is a type of algebra where the input and output values can only be true (1) or false (0).
  • A complete and completely distributive
    Boolean algebra is isomorphic to a CABA, whence there do exist free
    complete and completely distributive Boolean algebras.
  • However it would not be identical to our original Boolean algebra because now we find ∨ behaving the way ∧ used to do and vice versa.

There are two basic theorems of great importance in Boolean Algebra, which are De Morgan’s First Laws, and De Morgan’s Second Laws. In particular, the finitely many equations we have listed above suffice. We say that Boolean algebra is finitely axiomatizable or finitely based. The lines on the left of each gate represent input wires or ports.

Absorption Law

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. In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). We can simplify boolean algebra expressions by using the various theorems, laws, postulates, and properties.

The subset of
B consisting of the former is called an ultrafilter of B. 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.

What is Boolean Algebra laws?

This two-element algebra shows that a concrete Boolean algebra can be finite even when it consists of subsets of an infinite set. It can be seen that every field of subsets of X must contain the empty set and X. Hence no smaller example is possible, other than the degenerate algebra obtained by taking X to be empty so as to make the empty set and X coincide.

In Boolean logic, zero (0) represents false and one (1) represents true. In many applications, zero is interpreted as false and a non-zero value is interpreted as true. There are many other definitions of a Boolean algebra besides the
ones already encountered. As it would take us too far afield to develop the necessary background
for each, we leave a fuller understanding of these as a project to
pursue with the help of a suitable search engine. The category Bool of Boolean algebras has as objects all
Boolean algebras and as morphisms the homomorphisms between them. The first four pairs of axioms constitute a definition of a bounded lattice.

Boolean algebras

These operations furthermore satisfy
all the Boolean identities. Hence this set of sets forms a field
of sets in the above sense of Birkhoff and hence a Boolean algebra. Every BA \(A\) can be
embedded in a complete BA \(B\) in such a way that every element of
\(B\) is the least upper bound of a set of elements of \(A\). \(B\) is
unique up to \(A\)-isomorphism, and is called the completion of

In this way our first example can serve not only to illustrate
the concept but also to complete the definition of Boolean algebra
by defining Boolean logic. 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). The inverse of the boolean variable is called the complement of the variable. These operations have their own symbols and precedence and the table added below shows the symbol and the precedence of these operators.

A Boolean algebra (B,∨,∧,¬) is an algebra, that is,
a set and a list of operations, consisting of a nonempty set B, two
binary operations x∨y and x∧y, and a unary operation ¬x,
satisfying the equational laws of Boolean logic. In the Boolean Algebra, we have identity elements for both AND(.) and OR(+) operations. The identity law state that in boolean algebra we have such variables that on operating with AND and OR operation we get the same result, i.e. Variables used in Boolean algebra that store the logical value of 0 and 1 are called the boolean variables. The original application for Boolean operations was mathematical logic, where it combines the truth values, true or false, of individual formulas.

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.

T or 1 denotes ‘True’ & F or 0 denotes ‘False’ in the truth table. Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two values of fundamental importance to computer hardware, mathematical logic, and set theory. The set of finite and cofinite sets of integers, where a cofinite set is one omitting only finitely many integers. This is clearly closed under complement, and is closed under union because the union of a cofinite set with any set is cofinite, while the union of two finite sets is finite.

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