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Math2039 definitions

class test one - until integration

Injective f(a)=f(b) implies a=b
Surjective ∀b∈B ∃a∈A such that f(a)=b
Field F, a non empty set with 2 binary operations + and . such that (F,+) is an abelian group with identity 0, (F\{0},.) is an abelian group with identity 1 and ∀a,b,c∈F a(b+c)=ab+ac
Ordered Field relation < on F satisfying the following ∀a,b,c∈F (i) with a<b, a=b, or b<a (ii) a<b and b<c implies a<c (iii) a<b implies a+c<b+c (iv) a<b and 0<c implies ac<ab
Absolute Value of every element of an ordered field |x|= x if 0<x, -x if x<0, 0 if x=0
Completeness Axiom for an ordered field F we say F is complete if F=A∪B such that A∩B=∅ and ∀a∈A, ∀b∈B a<b implies ∃p∈F such that a≤p≤b
Archimedian Property ∀a∈ℝ, ∃n∈ℕ, such that 0<(1/n)<a
Supremum let ∅≠A⊆ℝ. sup(A)=s∈ℝ if (i) ∀a∈A, a≤s (ii) if ∃b∈ℝ such that ∀a∈A, a≤b then s≤b
Created by: meggle