Theorem 0.1 Division Algorithm
Let 
and 
be integers with 
. Then 
unique integers 
and 
with the property that 
, where 
.
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 
.
and integers and gcd(). Moreover, gcd() is the smallest positive integer of the form .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 .
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
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)
(
(reflexive) implies (symmetric) and imply (transitive)Partition
A partition of a set is a collection of nonempty disjoint subsets of whose unions is .
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
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 .
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 .
Kartensatzinfo:
Autor: Squiggleart
Oberthema: Mathematics
Thema: Abstract Algebra
Schule / Uni: West Chester University
Veröffentlicht: 25.05.2011
Tags: Algebra
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