Natural numbers

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• Distinct Natural Number Function

  For each m belonging to Natural numbers, there exists a natural number n such that s(n) = m. 2. For all m and n belonging to natural numbers, if m != n, then s(m) !

• Defining Certain Subsets of the Natural Numbers

  Now consider the L-structure that has as universe the set N of all the natural numbers (0, 1, 2, ...) and where the function symbol "+" is interpreted as the usual addition on the natural numbers.

• Properties of Natural Numbers

  236265 Properties of Natural Numbers Let m denote a natural number. Use any properties of the natural numbers to argue that m x m is not equal to m + m.

• Well-Ordering Axiom - Strong Induction

  Therefore, by strong induction, P(i) is true for all natural numbers i. That is, a non-empty subset of the natural numbers of any finite size has a smallest value.

• Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

  40010 Proofs : Pairwise Real Numbers, Natural and Irrational Numbers Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if λ1...λn are such real numbers that the equality ...

• Use of number properties

  Properties of numbers is discussed using their definitions. Commutative property, distributive property of rational numbers, natural numbers, irrational numbers is discussed.

• Numerical Concepts : Definitions

  given any two elements a, b, in some set (the integers, for example), closure means their sum is also in that set for instance, the (set of) natural numbers above is closed under addition : the sum of two natural numbers is always a natural number

• Sets : Venn Diagrams and Natural Numbers

  Venn diagrams and natural numbers are discussed.

• Continuity Proofs of Functions

  Since every subset of natural numbers is open in natural numbers, inverse image of an open set in R is open in natural numbers. Hence any function f mapped from natural numbers to R is continuous. 2.