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- Virginia
- Lynchburg College
- Mathematics
- Mathematics 405
- Cline
- Abstract Algebra

Lewis W.

Well Ordering Principle

Every nonempty set of positive integers contains a smallest number.

Division Algorithm

Let a and b be integers with b > 0. Then there exist unique integers a and r with the property that a = bq +r where 0<=r<=b

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Greatest Common Divisor/Relatively Prime Integers

The *greatest common divisor* of two nonzero integers a and b is the largest of all common divisors of a and b. We denote this integer by gcd(a,b). When gcd(a,b) = 1, we say a and b are *relatively prime*.

a mod n

Let *n *be a fixed positive integer. For any integer *a*, *a* mod *n* is the remainder upon dividing *a *by *n*.

Modular Equations

If *a *and *b *are integers and *n *is a positive integer, we write *a* = *b* mod n when *n *divides *a - b*.

First Principle of Mathematical Induction

Let S be a set of integers containing a. Suppose S has the property that whenever some integer n >= a belongs to S, then the integer n + 1 also belongs to S. Then, S contains every integer greater than or equal to a.

Second Principle of Mathematical Induction

Let S be a set of integers containing a. Suppose S has the property that n belongs to S whenever every integer less than n and greater than or equal to a belongs to S. Then, S contains every integer greater than or equal to a.

Equivalence Relation

An equivalence relation on a set S is a set R of ordered pairs of elements such that the reflexive, symmetric, and transitive properties hold.

Partition

A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S.

Function (Mapping)

A function k (or mapping) from a set A to a set B is a rule that assigns to each element a of A exactly one element b of B. (Definition Shortened)

Composition of Functions

let k: A-->B and j:B-->C. The composition jk is the mapping from A to C defined by (jk)(a)=j(k(a)) for all a in A.

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One-to-One Function

A function k from a set A to B is called one-to-one is k(a1) = k(a2) implies a1=a2.

Function from A onto B

A function k from a set A to a set B is said to be onto B if each element of B is the image of a least one element of A.

Binary Operation

Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G.

Group

Let G be a nonempty set together with a binary operation that assigns each ordered pair (a,b) of elements of G an element in G denoted by ab. We say G is a group under this operation if it is associative, has an identity, and inverses. (Shortened)

Abelian

If a group has the property that ab =ba for every pair of elements a and b, we say the group is Abelian.

Least Common Multiple

The least common multiple of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b. We will denote this integer by lcm(a,b).

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