ma103 definitions

7/30/2019 Ma103 Definitions

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Michaelmas Term

Pigeonhole

Principle

Let me be a number. Then for all : if there is aninjection from

Infinite sets A set which has an infinite number of elements. To prove,

use contradiction i.e. a bijection from a finite set from an

infinite set leads to a contradiciton

Composition of

functions

(gf)(x) = g(f(x))

Equivalence

Relation

It is an equivalence relation if it is

Reflexive Symmetric Transitive

Formally, a relation R on a set X is a subset of the Cartesian

product X XEquivalence Class Suppose R is an equivalence relation on a set X and for


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[] . Each [x]is a subset of X

Well ordering

principle of

integers

If S is a non empty set of integers with a lower bound, then S

has a least member.

Greatest commondivisor

Suppose a, b are two integers, at least one of which is not 0.The gcd(a,b) is the unique positive integer such that d

divides both a and b and d is greater than every other

common divisor of a and b. if c|a and c|b then c d

Euclidian

algorithm

Standard method for computing gcd(a,b)

The Fundamental

Theorem of

Arithmetic

Every integer n2 can be expressed as a product of one or

more prime numbers. Furthermore, there is essentially only

one such way of expressing n: the only way in which two

such expressions for n can differ is in the ordering of the

prime factors.Congruence

modulo m

aRb => m|(a-b). If aRb then we write

Congruence

classes modulo m

These are the equivalence classes of the above equivalence

relation, denoted [x]mInvertible

elements in Zm

An element x is invertible if there is some y such that xy = yx

= 1. If x is invertible, we can cancel it from modular

equations. X is invertible if and only if gcd(x, m) = 1 i.e. they

are coprime.

Rational

Numbers

Can be defined as an equivalence relation from Natural

numbers. (m, n)R(m, n) => mn = mn (which implies m/n =

m/n)Multiplication and addition of rationals can be defined as

operation on these equivalence classes.

Integers Can be defined as an equivalence relation from Natural

numbers

(a, b)R(c,d) => a +d = b + c (which implies a-b = c-d, but we

want to only use addition)

Real numbers They can be constructed, but is outside the scope of MA103.

Irrational

Numbers

Not rational it cannot be written as a/b where a,b are

natural numbers.

Countability ofRationals

Uses the cantor diagonal argument.

DeMoivres

Theorum

Z^n = r^n(cos n + isin n)

Eulers formula e^(i) = cos + i sin

Countable There is a bijection between the set and the Natural numbers

Complex

conjugate

a+bi conjugate: a-bi

Lent Term


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Upper Bound

Lower Bound

Supremum Least Upper Bound

Infimum Greatest Lower BoundMaximum Minimum Least upper

bound property

Any non-empty subset of that is bounded above, has asupremum.

Archimidean

Property

Interval A set consisting of all the real numbers between two given

real numbers, or of all the real numbers one side or the other

of a given real number. (9 different forms)

Absolute Value || Triangle

Inequality

| | | | | |

Symmetric

Absolute Value

| | | |

Sequence A sequence is, fundamentally, just a function f: Limit of a

sequence

| ||

Bounded

sequence

||

Monotonicallyincreasing

Sandwich

theorem

If Lim a = lim b and a


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[ ]] | [ ]]Group A group is a set G together with a law of composition (a,b)

a*b : G x G G which is Associative (G1), Contains an

Identity (G2) and each element has an inverse which is in the

group (G3).

Finite Group A group is a finite group if the set G has finite cardinality.The order of a finite group (G, *) is the cardinality of G. A

group is said to be an infinite group if it is not finite.

Subgroup A subset H of the set G is a subgroup if it is closed, it contains

the identity, and it has invesrses.

Abelian Group An abelian group has all the properties of a group, AND the

group operation is commutative.

GL(n,R) This is the group comprising of nxn matricies with real

entries.

The order of a

group

The order of a group is the cardinality of the set G on which

the law of composition acts. It is the size of the set S.Order of an

element of a

group

The least m such that am=e.

Finite order If there exists such an m [ord(a)=min{m|am=e}

Infinite order The negation of finite order

Cyclic Group A group is cyclic if there exists an element a in G such that G

= , the set generated by taking powers of a.

Homomorphism Let (G,*) and (G,*) be groups. A homomorphism is a

function: Isomorphism A homomorphism that is also bijective.

Coset Given a subgroup H of a group G, the relation R on G given by

aRb if b=a*h for some h in H. If a is in G, then the equivalence

class of a = {a*h|h is in H} is a left coset

Lagranges

theorem

Let H be a subgroup of a finite group G. Then the order of H

divides the order of G.

Vector Space A vector space is a set along with two functions, +:VxV V,

called vector addition, and another function, called scalar

multiplication, such that (V,+) is an abelian group. The

following hold:

1.v=v a(bv)=(ab)v (a+b)v=av+bv a(v+u)=av+au

Supace of a

vector space

Let U e a subset of V, U is a subspace if:

0 is in U v and n are in U then v+n is in U av is in U

Linear

Combination

v1,v2vn are in V and a1 a2an are in R then a1v1 +

a2v2.anvn is a linear combination of v1vn

always a finite set of vectors.Span Lin(empty set) = 0


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For all else, the linear span is All possible combinations of

vectors from S.

Linearly

dependent

Basis A set of vectors B is a basis of V if:

Lin(B)=V B is linearly independent

Dimension of V If there exists a basis of V with n elements, then n is the

dimension of V

Finite

Dimensional

Space

A space which has a basis with a finite number of elements

Infinite

Dimensional

Space

Not finite dimensional

LinearTransformation

T(u+v)=T(u)+T(v)T(au)=aT(u)

Kernel {u|T(u)=0}

Image Also the range, column space.

{v|there exists u with T(u)=v}

{v|T(u)=v}

{T(u)|u is in U}

Nullity Dim(ker)

Rank Dim(im)

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