Most of the sets dealt with in mathematics have an algebraic structure. That is, one or more rules of combination are defined between elements of the set. The most common examples of such sets are the various collections of numbers, such as the set of all integers,  the set of all real numbers, and the set of all complex numbers. There are four rules of combination for the real numbers: addition, subtraction,  multiplication and division. (The last rule is something of an exception,  in that division by zero is not permitted.) In the algebra of sets we observed two rules of combination, intersection and union.

Before an axiomatic definition of Boolean algebra can be given, it is necessary to discuss the nature of such rules of combination, which we will term binary operations. The definition which follows uses a symbol "∘" to stand for an arbitrary binary operation. Specific examples of ∘ would be symbols such as (.),(+)and (-).

DEFINITION. A binary operation ∘on a set M is a rule which assigns to each ordered pair (a, b) of elements of M a unique element c = a ∘ b in M.

EXAMPLE 1. The operation of subtraction is a binary operation on the set of all rational numbers (numbers of the form p/q, where p and q are integers and q is not zero) but is not a binary operation on the set of all positive integers.

For any two rational numbers A = p/q and B = r/s, the difference A - B is uniquely defined and is another rational number (ps - rq)/qs, hence (-) satisfies the conditions of the definition of a binary operation on the set of all rational numbers. However, the difference of two positive integers is not always a positive integer and hence (-) does not represent a binary operation on the set of positive integers.

While addition, multiplication, subtraction, and division are familiar examples of binary operations, the definition does not restrict the concept even to the extent that it have any intuitive or useful meaning whatsoever.

DEFINITION. A binary operation ∘ on a set of elements M is associative if and only if for every a, b, and c in M,

                                   a∘ (b∘ c)= (a∘ b) ∘ c.

DEFINITION. A binary operation a on a set M is commutative if and only if for every a and b in M,

                                       a∘b=b∘a.

DEFINITION. If ∘and * are two binary operations on the same set M, ∘ is distributive over * if and only if for every a, b. and c in M,

                             a∘ (b*c)=(a∘ b)* (a∘ c).

As examples, recall that in the algebra of sets, intersection and union were both commutative and associative, and each was distributive over the other. The distributive law for multiplication over addition is written as a(b + c) = ab + ac, and for addition over multiplication is written as a + bc= (a+b)(a+c).

In considering the set of all integers under the operation of addition. The number 0 stands out as different from all others because of the property that 0 + a = a for every integer a. This property is extremely important,  and algebraic systems which have such an element differ geratly from those which do not. The following definition gives a name to such elements.

DEFINITION. An element e in a class M is an identity for the binary operation = if and only if a ∘ e =e ∘a = a for every element a in M.

مواضيع ذات صلة

Zero Polynomial

Kronecker,s Polynomial Theorem

THE ALGEBRA OF SETS-The number of elements in a set

THE ALGEBRA OF SETS-Solution of equations

THE ALGEBRA OF SETS-Conditional equations

THE ALGEBRA OF SETS-Properties of set inclusion

THE ALGEBRA OF SETS- Expanding, factoring, and simplifying

THE ALGEBRA OF SETS-Fundamental laws

THE ALGEBRA OF SETS-Venn diagrams

THE ALGEBRA OF SETS-The combination of sets

THE ALGEBRA OF SETS-Element and set

THE ALGEBRA OF SETS-Introduction