mmw midterms

mathematics

study of numbers

pattern

an arrangement which helps observers anticipate what they might see or happens next

symmetry

something that repeats again and again

reflection

simplest kind of symmetry

radial

rotational symmetry around a fixed point known as the center

rosette patterns

taking motif or an element and rotating and/or reflecting that element

cyclic

only admits rotational symmetries

dihedral

admits both rotational symmetries and bilateral or reflectional symmetries

frieze

a basic motif repeats itself over and over in one direction

hop

the frieze pattern only admits a translational symmetry

step

the frieze pattern only admits a translational and glide symmetries

slide

the frieze pattern only admits translations and vertical reflections

spinning hop

the frieze pattern only admits the translations and 180 rotations

spinning siddle

the frieze pattern only admits translations, vertical reflections, rotations, and glide reflections

jump

the frieze pattern only admits translations, a horizontal reflection, and glide reflection

spinning jump

the frieze pattern admits translations, vertical reflections, horizontal reflections, rotations, and glide reflections

wallpaper patterns

pattern with translation symmetry in two directions

tesselations

repeating pattern of figures that covers a plane with no gaps or overlaps

Pisano

means from Piso

Fibonacci

which means “Son of Bonacci”

recursive definition

starts by defining some initial values defines the next term as a function of

golden ratio

as n gets larger and larger, the ratio gets closer and closer to a value

binet formula

we can write the nth fibonacci number explicitly using the formula

mathematical modelling

see the inputs to events and their most likely outcomes

control theory

a field of applied mathematics that is relevant to the control of certain physical processes and systems

indispensable tool

needed by all people in honing their logical thinking and reasoning

integers

whole numbers including zero and negative values

rational numbers

repeating or terminating decimals

irrational

non-terminating, non-repeating decimals

real numbers

the set containing all rational and irrational numbers

sequence

is an ordered list of numbers that follow a specific rule or pattern

explicit formula

expresses a”n” directly in terms of n

recursive formula

defines a'n’ using preceding terms

arithmetic sequence

sequence where each term increases by a constant difference

geometric sequence

sequence where each term is multiplied by a constant ratio

alternating sequences & negative numbers

sequences using (-1)n to flip signs between negative and positive terms

harmonic sequences & fractions

a sequence formed by taking the reciprocals of an arithmetic progression

convergent sequence

sequence terms approach a fixed real number L as n

Divergent sequence

sequence terms grow indefinitely or oscillate without approaching a single value

communication

process of exchanging information

language

a system of conventional spoken, manual, or written symbols which human beings express themselves

symbols

arbitrary marks, signs, or gestures that represent concepts, objects, or relationships beyond their literal form

english noun

person, place or thing

math expression

name given to a mathematical object of interest, containing no complete thought

english sentence

a complete thought with a subject and verb

math sentence

a complete thought stating a relation, containing a truth value

precise

concise

powerful

core characteristics of mathematical language

expression

a correct arrangement of mathematical symbols used to represent symbols used to represent a mathematical object of interest

sets

well-defined collection of distinct objects

The foundational collections that define the domain of objects being studied

relation

a general connection or association between elements of two sets

functions

special type of relation between two sets where every input element from the domain is mapped to exactly one output element in the codomain

binary operations

rule that takes two elements from a set and combines them to produce a single element from the same set

unit set

contains only one element

empty or null set

has no element

finite set

the elements in a given set is countable

infinite set

elements in a given set has no end or not countable

cardinal number

numbers that used to measure the number of elements in a given set

counting rule

only distinct elements are counted

duplicate elements within set notation are ignored

equal set

two sets are equal if and only if they contain the exact same elements regardless of the order they are listed

equivalent set

two sets are equivalent if they have the same number of elements

universal set

the set of all elements under discussion

joint set

if and only if they have common

disjoint set

mutually exclusive or if they dont have common elements

roster/tabular method

lists every individual element of the set explicitly inside curly braces

rule/set-builder method

specifies a set by stating a common property or condition that all elements must satisfy, using vertical bar notation

complement

is the subset of all elements of S that are not in A

intersection

the event containing all elements that are common to A and B

union

the event containing all the elements that belong to A or B or both

mutually exclusive/disjoint

if A and B have no elements in common

commutative law

order does not matter

associative law

grouping does not change the result

identity law

unifying any set with an empty set leaves it unchanged

domination/universal law

unifying any set with the universal set gives the universal set

mutually exclusive sets

they share no elements in common

partition

divides a non-empty set into non-overlapping sub-groups

non-empty subsets

none of the subsets in the partition are empty

pairwise disjoint

no two subsets share any elements in common. their intersection is always empty

exhaustive union

the union of all subsets in the partition recreates the original set S entirely

partition

a collection of sets

De Morgan’s Laws

complement operations interact with unions and intersections

non-commutative

order does matter

relation to complement

the difference is equivalent to the intersection of A with the complement of B

self difference

subtracting a set from itself leaves an empty set

ordered pairs

the first element comes from set A and the second element comes from set B

relation

a fundamental concept that describes a connection or association between elements of sets

domain

all x-values (inputs)/the set of all first components

codomain

all y-values (outputs)/the set of all second components

range

set where all outputs for the established relationship belongs

range

the elements actually mapped to by domain inputs

mapping diagram

uses two ovals to represent domain A and codomain B, drawing directed arrows from each input x to its corresponding output y

matrix representation

main diagonal of the relation’s adjacency matrix must consist entirely of 1s

directed graph

every node/vertex must have a self-loop (an arrow starting and ending at the same node)

symmetric relation

whenever an element a is related b, then b must also be related to a

reflexive

every element is related to itself

transitive

if a is related to b and b to c, then a is related to c

many-to-one relation

several elements from set A relate to one element in set B

injective

every input produces a unique output

non-injective

two distinct inputs yield the exact same output

surjective

every element in the codomain B is mapped to by at least one element in domain A

Bijective

both injective and surjective.