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# Math2039 definitions

### class test one - until integration

Question | Answer |
---|---|

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 |

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meggle