Examples
 The simplest nontrivial Boolean algebra, the twoelement Boolean algebra, has only two elements, 0 and 1, and is defined by the rules:




 It has applications in logic, interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.

 The twoelement Boolean algebra is also used for circuit design in electrical engineering; here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same inputoutput behavior. Furthermore, every possible inputoutput behavior can be modeled by a suitable Boolean expression.

 The twoelement Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the twoelement Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
 (a ∨ b) ∧ (¬a ∨ c) ∧ (b ∨ c) ≡ (a ∨ b) ∧ (¬a ∨ c)
 (a ∧ b) ∨ (¬a ∧ c) ∨ (b ∧ c) ≡ (a ∧ b) ∨ (¬a ∧ c)
 The twoelement Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the twoelement Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
 The power set (set of all subsets) of any given nonempty set S forms a Boolean algebra with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the empty set and the largest element 1 is the set S itself.

 After the twoelement Boolean algebra, the simplest Boolean algebra is that defined by the power set of two atoms:



 The set of all subsets of S that are either finite or cofinite is a Boolean algebra.
 Starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo tautology). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the twoelement Boolean algebra.
 Given any linearly ordered set L with a least element, the interval algebra is the smallest algebra of subsets of L containing all of the halfopen intervals [a, b) such that a is in L and b is either in L or equal to ∞. Interval algebras are useful in the study of LindenbaumTarski algebras; every countable Boolean algebra is isomorphic to an interval algebra.
 For any natural number n, the set of all positive divisors of n forms a distributive lattice if we write a ≤ b for a  b. This lattice is a Boolean algebra if and only if n is squarefree. The smallest element 0 of this Boolean algebra is the natural number 1; the largest element 1 of this Boolean algebra is the natural number n.
 Other examples of Boolean algebras arise from topological spaces: if X is a topological space, then the collection of all subsets of X which are both open and closed forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).
 If R is an arbitrary ring and we define the set of central idempotents by
A = { e ∈ R : e2 = e, ex = xe, ∀x ∈ R }
then the set A becomes a Boolean algebra with the operations e ∨ f := e + f  ef and e ∧ f := ef.
Read more about this topic: Boolean Algebra (structure)
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“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often illapplied examples of his wrath.”
—G.C. (Georg Christoph)
“No rules exist, and examples are simply lifesavers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)