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## Algebra 1 (10 Cards)

Theorem 0.1 Division Algorithm
Let and be integers with . Then unique integers and with the property that , where .
Theorem 0.2 GCD is a Linear Combination
For any nonzero integers and integers and gcd( ) . Moreover, gcd( ) is the smallest positive integer of the form .
Euclid's Lemma
if p is prime then p|ab implies p|a and/or p|b.
Theorem 0.3 Fundamental Theorem of Arithmetic
Every integer greater than 1 is a prime or the unique product of primes.
Theorem 0.4 First Principle of Mathematical Induction
Let be a set of integers containing . Suppose has the property that whenever some integer , then the integer . Then, contains every integer greater than or equal to .
Theorem 0.5 Second Principle of Mathematical Induction
Let be a set of integers containing . Suppose has the property that whenever every integer less than and greater than or equal to belongs to . Then, contains every integer greater than or equal to Equivalence Relation
An equivelence relation on a set S is a set R of ordered pairs ot elements of S such that:
( (reflexive) implies (symmetric) and imply (transitive)
Partition
A partition of a set is a collection of nonempty disjoint subsets of whose unions is .
Theorem 0.6 Equivalence Classes Partition
The equivalence classes of an equivalence relation on a set constitute a partition of . Conversely, for any partition of , there is an equivalence relation on whose equivalence classes are the elements of Function (Mapping)
A function (or mapping) from a set to a set is a rule that assigns to each element of exactly one element of . The set is called the domain of and is called the range of . If assigns to , then is called the image of under . The subset of comprising all the images of elements of is called the image of A under . Flashcard set info:
Author: Squiggleart
Main topic: Mathematics
Topic: Abstract Algebra
School / Univ.: West Chester University
Published: 25.05.2011
Tags: Algebra

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