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Identity and Inverse

Labels: Group Theory

Identity: A composition

in a set

is said to possesses of an identity if there exists an element

such that

Moreover, the element e, if it exists is called an identity element and the algebraic structure

is said to have an identity element with respect to

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The concept of binary operation on a set is a generalization of the standard operations like addition and multiplication on the set of numbers. For instance we know that the operation of addition (+) gives for ally two natural numbers m,n another natural number m+n, similarly the multiplication operation gives for the pair m,n the number m,n in N again. These types of operations arc found to exist in many other sets. Thus we give the following definition.

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Archimedean Property of Real Number

Labels: Real Analysis

Let

x

be any real number. Then there exists a natural number

n

such that

n>x

This theorem is known as the Archimedean property of real numbers. It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a consequence of the least upper bound property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus).

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Identity and Inverse

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Archimedean Property of Real Number