Plane Geometry - Bruce E. Shapiro

SECTION 6. REAL NUMBERS 27Definition 6.6 (Upper Bound) A number M is called an upper boundfor a set A if ∀x ∈ A, x ≤ M.Definition 6.7 (Least Upper Bound) A number m is called a leastupper bound for a set A if for all upper bounds M of A, m ≤ M, and wewrite m = lubA.Axiom 6.8 (Least Upper Bound Axiom) Every bounded non-emptysubset of the real numbers has a least upper bound.The following property expresses the notionthat you can fill up a bucket with spoonfuls ofwater. We will accept it as an axiom althoughit fact it can be derived from the Least UpperBound Axiom.Axiom 6.9 (Archimedian Property) IfM > 0, ɛ > 0 are both real numbers than thereexists a postive integer n such that nɛ > M.Definition 6.10 (Function) A function f isa rule that assigns to each element a ∈ A anelement b = f(a) ∈ B. We call A the domainof f, and we call the subset of B to whichelements of A are mapped by f the range off. We write f : A ↦→ B.Definition 6.11 A function f : A ↦→ B isone-to-one (sometimes 1-1 or (1:1)) if a 1 ≠a 2 ⇒ f(a 1 ) ≠ f(a 2 )Babylonian clay tablet YBC7289 (c 1800-1600 BC) showing√2 ≈ 1 +2160 + 5160 2 + 1060 3(figure Bill Casselmanhttp://www.math.ubc.ca/~cass/Euclid/ybc/ybc.html.)Definition 6.12 A function f : A ↦→ B is onto if (∀b ∈ B)(∃a ∈ A) suchthat b = f(a).Definition 6.13 A function f : A ↦→ B that is 1-1 and onto is called aone-to-one-correspondence.Definition 6.14 A function f(x) is continuous on an interval (a, b) if forevery ɛ > 0 there exists a δ > 0 such that whenever |x − y| < δ, x, y ∈ (a, b),then |f(x) − f(y)| < ɛ.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.