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parts of a =bq ; b|a

aZ =

Well-Ordering Principle

Divison Algorithm

a is a multiple, b is a divsor (factor), q is some integer

= { m ∈ Z | m=aq for some q ∈ Z }

Every nonempty set of natural numbers contains a smallest ele…

For any a, b, there exists q, r such that a=bq+r

parts of a =bq ; b|a

a is a multiple, b is a divsor (factor), q is some integer

aZ =

= { m ∈ Z | m=aq for some q ∈ Z }

Well Ordering Principle

Division Algorithm

GCD is a Linear Combination

GCD corollary

Every nonempty set of positive integers contains a smallest m…

Let a and b be integers with b>0. Then there exists unique in…

For any nonzero integers a and b, there exists integers s and…

If a and b are relatively prime, then there exists integers s…

Well Ordering Principle

Every nonempty set of positive integers contains a smallest m…

Division Algorithm

Let a and b be integers with b>0. Then there exists unique in…

Binary Operations

Operation Properties

Group

Abelian

Defined over all ordered pairs;... Unique result;... If operands ar…

Commutative;... Associative;... Identity element;... Inverse

A set G and operation * that is associative, has a neutral el…

Group G with operation * is commutative.

Binary Operations

Defined over all ordered pairs;... Unique result;... If operands ar…

Operation Properties

Commutative;... Associative;... Identity element;... Inverse

Well-Ordering Axiom

Divisbility

Greatest Common Divisor (gcd)

Relatively Prime

Every nonempty subset of the set of nonnegative integers cont…

Let a and b be integers with b =/ 0. We say b|a or b is a fac…

Let a and b be integers, not both 0. The gcd of a and b is th…

Two integers whose greatest common divisor is 1

Well-Ordering Axiom

Every nonempty subset of the set of nonnegative integers cont…

Divisbility

Let a and b be integers with b =/ 0. We say b|a or b is a fac…

Associativity

The Well-Ordering Priciple

injectivity

surjectivity

addition and multiplication of real numbers is... (x+y)+z = x +…

Let S be a nonempty subset of the natural numbers. Then with…

S,T sets, f:S to T f is injective (or 1 to 1) if f(s_1)=f(s_2…

S,T sets, f:S to T f is surjective (or onto) if for all t in…

Associativity

addition and multiplication of real numbers is... (x+y)+z = x +…

The Well-Ordering Priciple

Let S be a nonempty subset of the natural numbers. Then with…

Proper Subset

Cartesian Product

Domain and Range of A-> B

Surjective or Onto

B is a subset of A but they are NOT equal

AxB {(a,b) :a elem of A/ b elem of B}

Domain A... Range B

if f: A->B is a map and the image of f is B then f is surject…

Proper Subset

B is a subset of A but they are NOT equal

Cartesian Product

AxB {(a,b) :a elem of A/ b elem of B}

Element

Empty Set

Well-Defined

Binary Operation

A set S is made up of elements, and if a is one of these elem…

There is exactly one set with no elements. It is the empty se…

A set is well-defined, meaning that if S is a set and a is so…

A binary operation * on a set S is a function mapping S x S i…

Element

A set S is made up of elements, and if a is one of these elem…

Empty Set

There is exactly one set with no elements. It is the empty se…

equivalence relation

Greatest common divisor on z

isomorphism between groups

group

a relation that satisfies three properties for all x,y,z in s…

let r and s be two positive integers. the positive generator…

a function between two groups that sets up a one-to-one corre…

a set, closed under binary operation that is associative, ide…

equivalence relation

a relation that satisfies three properties for all x,y,z in s…

Greatest common divisor on z

let r and s be two positive integers. the positive generator…

closed under the operation

closed under taking the inverses

Subgroup

Order of a Group

if ab is in H and whenever a and b are in H

a inverse is in H whenever a is in H

if the subset H of group G is itself a group under the operat…

the number of elements of a group (infinite or finite) is cal…

closed under the operation

if ab is in H and whenever a and b are in H

closed under taking the inverses

a inverse is in H whenever a is in H

conditions for a group

dihedral group

elements a, b ∈ G are conjugate if...

conditions for an equivalence relation

binary operation, identity element, inverses, associativity

set of symmetries on a regular n-gon

there exists c ∈ G so that c⁻¹ * a * c = b.

reflexive, symmetric, transitive

conditions for a group

binary operation, identity element, inverses, associativity

dihedral group

set of symmetries on a regular n-gon

GCD as a linear combination

One-Step Subgroup Test

Two-Step subgroup test

Finite subgroup Test

Let a and b be positive integers. Then there exist integers s…

Let G be a group and H a non-empty subset of G. If ab-1 whene…

If H is a non-empty subset of G and whenever a,b, E H we know…

If H is a finte, non=empty subset of a group G, and ab E G wh…

GCD as a linear combination

Let a and b be positive integers. Then there exist integers s…

One-Step Subgroup Test

Let G be a group and H a non-empty subset of G. If ab-1 whene…

Binary Operation

Identity Element

Inverse Element

Group

A fcn *: G×G ⇨ G is called a binary operation on G (which mea…

There exists e ∈ G such that ... e**a = a**e... for every a ∈ G

For all a ∈ G there exists b ∈ G such that:... a**b = b**a = e... b…

A set G with a binary operation * is a group iff the followin…

Binary Operation

A fcn *: G×G ⇨ G is called a binary operation on G (which mea…

Identity Element

There exists e ∈ G such that ... e**a = a**e... for every a ∈ G

Well Ordering Principle

Division Algorithm

Greatest Common Divisor / Relative Pri…

GCD is a Linear Cmbination

Every nonempty set of positive integers contains a smallest m…

Let a and b be integers with b > 0. Then there exist unique i…

The greatest common divisor of two nonzero integers a and b i…

For any nonzero integers a and b, there exists integers s and…

Well Ordering Principle

Every nonempty set of positive integers contains a smallest m…

Division Algorithm

Let a and b be integers with b > 0. Then there exist unique i…

Groups

Binary operation

Commutative

Associative

An abstraction of the integers under addition

Let S be a set. A binary operation ** is a function ** : SxS -> S

Let ** be a binary operation on S. Then ** is commutative if fo…

Let ** be a binary operation on S. Then ** is associative if fo…

Groups

An abstraction of the integers under addition

Binary operation

Let S be a set. A binary operation ** is a function ** : SxS -> S

Division algorithm

GCD is a linear combination

fundamental theorem of Arithmetic

first principle of mathematical induct…

let a and b be integers with b greater than 0 then there exis…

For any nonzero integers a and b, there exist integers s and…

every positive integer n>1 is either a prime or a product of…

If (i) P(a) is true for the starting point a which is a posit…

Division algorithm

let a and b be integers with b greater than 0 then there exis…

GCD is a linear combination

For any nonzero integers a and b, there exist integers s and…

Well-ordering Principle

Greatest common divisor of a and b

THM: gcd(a,b)= MIN{

A partition of a set S is

Every non empty set of positive integers has a least element.

The greatest integer d such that d divides a and d divides b

ax+by such that x,y in Z and ax+by>0

collection of nonempty disjoint subsets of S whose union is S

Well-ordering Principle

Every non empty set of positive integers has a least element.

Greatest common divisor of a and b

The greatest integer d such that d divides a and d divides b

Element

Empty Set

Well-Defined

Binary Operation

A set S is made up of elements, and if a is one of these elem…

There is exactly one set with no elements. It is the empty se…

A set is well-defined, meaning that if S is a set and a is so…

A binary operation * on a set S is a function mapping S x S i…

Element

A set S is made up of elements, and if a is one of these elem…

Empty Set

There is exactly one set with no elements. It is the empty se…

Group

dihedral group

abelian/commutative group

order

set G with a binary operation '.': GxG→G that satisfies:... 1. (…

(Dn) group of symmetries of the regualr n-gon

if a₀b=b₀a for all a,b∈G

number of elements in the group... |Z4|=4

Group

set G with a binary operation '.': GxG→G that satisfies:... 1. (…

dihedral group

(Dn) group of symmetries of the regualr n-gon

Cayley's Theorem

Normal Subgroup Test

Cauchy's Theorem for Finite Abelian Gr…

|A|

Every group G is isomorphic to a (sub)group of permutations o…

A subgroup H of G is normal in G if and only if xHx⁻¹⊆H for a…

Let G be a finite Abelian group whose order is divisible by a…

The cardinality of A, the number of elements in A.

Cayley's Theorem

Every group G is isomorphic to a (sub)group of permutations o…

Normal Subgroup Test

A subgroup H of G is normal in G if and only if xHx⁻¹⊆H for a…

DEFINITION: Set equivalence (A = B)

DEFINITION: Subset (A ⊆ B)

DEFINITION: Empty set (Ø)

DEFINITION: Power set

Two sets, A and B, are equal (A=B) if and only if they contai…

For all a∈A, we also have a∈B.

The set that does not contain any elements. It is a subset of…

The set of subsets.

DEFINITION: Set equivalence (A = B)

Two sets, A and B, are equal (A=B) if and only if they contai…

DEFINITION: Subset (A ⊆ B)

For all a∈A, we also have a∈B.

factor group(G/H)

automorphism

simple

maximal normal subgroup

If φ G -> G' be a group homo with Kerφ = H, then the cosets f…

an isomorphism φ:G->G of a group with itself.

nontrivial group that has no proper nontrivial normal subgrou…

is a subgroup M<G M≠G s.t. There does not exist a proper norm…

factor group(G/H)

If φ G -> G' be a group homo with Kerφ = H, then the cosets f…

automorphism

an isomorphism φ:G->G of a group with itself.

Commutative

Identity element

Inverse

Operation

Independent of order ab=ba

a**e=a and e**a=a for every a in A

a**a^(-1)=e and a^(-1)**a=e for every a in A

Let A be any set. An operation on A has the following aspects…

Commutative

Independent of order ab=ba

Identity element

a**e=a and e**a=a for every a in A

Definition: Congruence modulo m

What is the set of integers to which a…

Definition: Least non-negative residue

Proof: Least non-negative residue

Two integers a and b are congruent modulo m if if m divides a…

a ≡ b (mod m) for all b s.t. b = a + km, where k is some scal…

Let m be a natural number > 1. Every m ∈ N is congruent modul…

This can be proved via application of the division theorem. C…

Definition: Congruence modulo m

Two integers a and b are congruent modulo m if if m divides a…

What is the set of integers to which a…

a ≡ b (mod m) for all b s.t. b = a + km, where k is some scal…

Cardinality

Cartesian Product

Function

Domain

the number of elements in a set. also called order. |A| of a…

AxB of two sets A and B is an ordered pair (x,y) with x∈A, y∈…

f:A→B a function takes elements from A and maps them to B. *m…

the set A from a function

Cardinality

the number of elements in a set. also called order. |A| of a…

Cartesian Product

AxB of two sets A and B is an ordered pair (x,y) with x∈A, y∈…

Subgroup

Centralizer

Center

Normalizer

Let G be a group. The subset H of G is a subgroup if H is non…

CG(A) = {g∈G | gag^-1=a for all a∈A}... CG(A) is the set of all…

Z(G) = {g∈G | gx=xg for all x∈G}... Z(G) is the set of elements…

NG(A) = {g∈G | gAg^-1 = A}... (G commutes with the set A But not…

Subgroup

Let G be a group. The subset H of G is a subgroup if H is non…

Centralizer

CG(A) = {g∈G | gag^-1=a for all a∈A}... CG(A) is the set of all…

Direct Product

Fundamental Theorem of finitely Genera…

Ring

Ring with identity

a direct product of Groups G and H is G x H = {(g,h) | g∈G, h…

Let G be a finitely generated abelian group, then G≅ℤ^r x ℤn1…

A ring R is a set together with 2 binary operations, + and *,…

a ring in which there is a multiplicative identity, 1R

Direct Product

a direct product of Groups G and H is G x H = {(g,h) | g∈G, h…

Fundamental Theorem of finitely Genera…

Let G be a finitely generated abelian group, then G≅ℤ^r x ℤn1…

Ring

Ring with Unity

Commutative Ring

Zero-Divisor

A nonempty set R is called a ring if R is equipped with 2 bin…

If there is an element 1R∈R s.t. 1R×a=a×1R=a ∀a∈R and 1≠0... (id…

If a×b=b×a ∀a,b∈R

If R is a ring, then a∈R is a zero-divisor if there exists an…

Ring

A nonempty set R is called a ring if R is equipped with 2 bin…

Ring with Unity

If there is an element 1R∈R s.t. 1R×a=a×1R=a ∀a∈R and 1≠0... (id…

Induction

What is a binary operation?

Is... S = ℝ, ... Φ as +... a binary operation?

Is... S = ℝ, ... Φ as ÷... a binary operation?

define p(n).... step 1: p(1) true?... step 2: Assume p(k) is true... s…

A binary operation on a set S is a map Φ:... S x S → S

Yes

No... Because:... 1 Φ 0 would not be defined

Induction

define p(n).... step 1: p(1) true?... step 2: Assume p(k) is true... s…

What is a binary operation?

A binary operation on a set S is a map Φ:... S x S → S

Binary Operation

Group

Order of a Group

Order of an Element

A function that assigns each ordered pair in G another elemen…

A set G with binary operation. If associativity, inverses and…

The number of elements in a group.

The smallest possible integer, n, such that g^n=e. If not n e…

Binary Operation

A function that assigns each ordered pair in G another elemen…

Group

A set G with binary operation. If associativity, inverses and…

Binary Operation

Group

Abelian

Euclidean Algorithm

A rule that takes in ordered pairs (a,b) from S×S and assigns…

a set G along with a binary operation that satisfies the foll…

A group with the property that a₀b=b₀a for all a,b ∈ G

a=bq+r

Binary Operation

A rule that takes in ordered pairs (a,b) from S×S and assigns…

Group

a set G along with a binary operation that satisfies the foll…

Ring

Unity

Unit

Thm 12.1 - 6 properties of multiplicat…

A ring R is a set with two binary operations, addition (a+b)…

A nonzero element that is an identity under multiplication.

A nonzero element of a commutative ring with a multiplicative…

1. a0 = 0a = 0 ... 2. a(-b) = (-a)b = -(ab)... 3. (-a)(-b) = ab... 4.…

Ring

A ring R is a set with two binary operations, addition (a+b)…

Unity

A nonzero element that is an identity under multiplication.

Cosets

Poperties of cosets

Lagrange's Thm

Index

G is a group, H c G... For all a in G, define aH = {ah | h E H}…

a E aH... AH = H iff a E H... AH = bH or aH (Union) bH = the empty…

If G is a finite group and H < G, then |H| divides |G|.... Order…

The index of H in G, written |G:H| is the number of left or r…

Cosets

G is a group, H c G... For all a in G, define aH = {ah | h E H}…

Poperties of cosets

a E aH... AH = H iff a E H... AH = bH or aH (Union) bH = the empty…

Generator

Cyclic Group

Centralizer

Subgroup

The element in the group whose order has the same number of e…

Has a generator

C(a) is the set of all elements in G in that commute with a

Satisfies Closure and Inverse rules

Generator

The element in the group whose order has the same number of e…

Cyclic Group

Has a generator

Let a, b ∈ Z such that a ≠ 0. Define a…

Define the order relation < on Z.

Let a, b ∈ Z such that a, b ≠ 0. Defin…

Let a, b ∈ Z and n ∈ N. Define a ≡ b.

Let a, b ∈ Z such that a ≠ 0. We say that a divided b, (or b…

Let a, b ∈ Z. We say that a is less than b, denoted a < b, if…

Let a, b ∈ Z such that a ≠ 0 and b ≠ 0. Then the greatest com…

Let n ∈ N. We say that an integer x is congruent to integer y…

Let a, b ∈ Z such that a ≠ 0. Define a…

Let a, b ∈ Z such that a ≠ 0. We say that a divided b, (or b…

Define the order relation < on Z.

Let a, b ∈ Z. We say that a is less than b, denoted a < b, if…

Greatest Common Divisor

Least Common Multiple

Example of Modular Arithmetic

Equivalence Relation

The ... of two nonzero integers a and b is the largest of all…

The ... of two nonzero integers a and b is the smallest posit…

a=bn+r then r=a mod n

...on a set S is a set R of ordered pairs of elements of S su…

Greatest Common Divisor

The ... of two nonzero integers a and b is the largest of all…

Least Common Multiple

The ... of two nonzero integers a and b is the smallest posit…

Well Ordering Principle

Division Algorithm

Greatest Common Divisor

a is congruent to b modulo n

Any nonempty set of positive integers contains a smallest mem…

Let a and b be integers with b>0. Then there exists unique in…

The ____ of two nonzero integers a and b is the largest of al…

Let a and b be integers and n a positive integer. When n divi…

Well Ordering Principle

Any nonempty set of positive integers contains a smallest mem…

Division Algorithm

Let a and b be integers with b>0. Then there exists unique in…

Zero divisor/ divisor of zero

Division Ring

Ring with Unity

Commutative Ring

A non-zero element a in a ring R provided ab=0 for some non-z…

if every non-zero element of R has a multiplicative inverses

If R had a multiplicative inverse of 1

If *is commutative

Zero divisor/ divisor of zero

A non-zero element a in a ring R provided ab=0 for some non-z…

Division Ring

if every non-zero element of R has a multiplicative inverses

Well-Ordered Principle

Lagrange

Fundamental Theorem of Finite Abelian…

Fundamental Theorem of Finitely Genera…

Every nonempty subset of the natural numbers is well ordered.

Let G be a finite group and let H be a subgroup of G. Then [G…

Every finite abelian group G is isomorphic to a direct produc…

Every finitely generated abelian group G is isomorphic to a d…

Well-Ordered Principle

Every nonempty subset of the natural numbers is well ordered.

Lagrange

Let G be a finite group and let H be a subgroup of G. Then [G…

Cartesian Product, S×T

a divides b

Binary Operation

Commutative Binary Operation

S×T={ (s,t) | s∈S, t∈T}

a|b, or b=ac, c integer

*: S×S → S, where (S₁, S₂) → S₁*S₂... ex. when S is all Real Num…

a*b=b*a for all a,b ∈ S... ex. × is commutative in all Real Numb…

Cartesian Product, S×T

S×T={ (s,t) | s∈S, t∈T}

a divides b

a|b, or b=ac, c integer

What is a ring?

What is a commutative ring?

What is a multiplicative unit?

What is a ring of polynomials?

A ring (R,+,×) is a set R with two binary operations + and ×…

A ring in which the multiplication is commutative.

Let (R,+,×) be a ring. R has a multiplicative unit (denoted b…

Given R with unit 1, the ring of polynomials over R (R[x]) is…

What is a ring?

A ring (R,+,×) is a set R with two binary operations + and ×…

What is a commutative ring?

A ring in which the multiplication is commutative.