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Archimedean Property of Real Number

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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Vector Space (Theorem-5)

Labels: Linear Algebra, Vector Space Theorem

Statement:- The necessary & sufficient conditions for a non-empty subset W of a vector spaceV(F) to be a subspace of V are

( 1) α∊W, β∊W ⇨ α-β∊W

(2) a∊F, α∊W ⇨ aα∊W

Proof:- Necessary Condition:-

Let V be a vector space over the field F and W be its subspace.

∴ W will be a vector space over the same field F.

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Field & Integral Domain

Labels: Abstract Algebra, Field

Integral domain:-

1- A commutative ring with unity element and without zero divisors is called an integral domain.

2- An algebraic system (D,+,.) where D is a non-empty set with two binary compositions to be denoted by addition and multiplication is called an integral domain if following axioms are satisfied-

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