How can we help?

You can also find more resources in our Help Center.

fundamental Counting Principle

Addition Principle

recursive definition

Fibonacci sequence

if one event can have x possible different outcomes, and afte…

address situations in which two different tasks are completed…

whenever one element of a sequence is defined in terms of a p…

0,1,1,2,3,5,8,13,21

fundamental Counting Principle

if one event can have x possible different outcomes, and afte…

Addition Principle

address situations in which two different tasks are completed…

statement

negation of ¬p

conjunction p^q

conjunction

an expression which is either true or false

p... and goes to or... and p goes to ¬p... MAKES P NOT TRUE

AND... "p and q"... both HAVE to be TRUE

p^q

statement

an expression which is either true or false

negation of ¬p

p... and goes to or... and p goes to ¬p... MAKES P NOT TRUE

statement

negation of ¬p

conjunction p^q

conjunction

an expression which is either true or false

p... and goes to or... and p goes to ¬p... MAKES P NOT TRUE

AND... "p and q"... both HAVE to be TRUE

p^q

statement

an expression which is either true or false

negation of ¬p

p... and goes to or... and p goes to ¬p... MAKES P NOT TRUE

fundamental Counting Principle

Addition Principle

recursive definition

Fibonacci sequence

if one event can have x possible different outcomes, and afte…

address situations in which two different tasks are completed…

whenever one element of a sequence is defined in terms of a p…

0,1,1,2,3,5,8,13,21

fundamental Counting Principle

if one event can have x possible different outcomes, and afte…

Addition Principle

address situations in which two different tasks are completed…

What is a function?

Why is the codomain and domain necessa…

How is the vertical line test related…

What are injective functions?

a function is simply a relation between a set of inputs and a…

let f be the function from x maps to x^2 -- why is this an in…

Can test whether or not each x input maps to exactly one y in…

The function is injective (one-to-one) if every element of th…

What is a function?

a function is simply a relation between a set of inputs and a…

Why is the codomain and domain necessa…

let f be the function from x maps to x^2 -- why is this an in…

Bipartite Graph

Circuit

Complement Graph

Complete Graph

A graph whose vertices can be divided into 2 sets such that n…

A walk that begins and ends at the same vertex, and has no re…

A graph with the same # of vertices as G but has an edge betw…

A simple graph in which each pair of vertices is joined by an…

Bipartite Graph

A graph whose vertices can be divided into 2 sets such that n…

Circuit

A walk that begins and ends at the same vertex, and has no re…

Permutation is how many arrangements a…

1

A traveler can choose from three airli…

Sequence

What is the difference between permutation and combination?

Compute ₅P₀

Multiplication Principle

An ordered list of numbers

Permutation is how many arrangements a…

What is the difference between permutation and combination?

1

Compute ₅P₀

Proposition

Propositional variables (or statement…

Compound propositions

p → q (Conditional Statement)

A proposition is a declarative sentence (that is, a sentence…

We use letters to denote propositional variables (or statemen…

The propositions formed from existing propositions using logi…

False only when p is True and q is False.... (Or)... Always True wh…

Proposition

A proposition is a declarative sentence (that is, a sentence…

Propositional variables (or statement…

We use letters to denote propositional variables (or statemen…

composite

prime

substituted

predicate

合成关系, 合成数;... An integer n is prime if, and only if, n>1 and for…

素数, 质数;

vicarious;... When an element in the domain of the variable of a…

a sentence that contains a ﬁnite number of variables and beco…

composite

合成关系, 合成数;... An integer n is prime if, and only if, n>1 and for…

prime

素数, 质数;

What is a digraph?... [Digraphs]

What is a path?... [Digraphs]

What is a simple path?... [Digraphs]

What is a cycle?... [Digraphs]

Directed Graph... Binary relation, R on A: R ⊆ A x A with direc…

Sequence of vertices connected with edges from vertex A to ve…

No repeated vertices

A path that starts and ends on the same vertice

What is a digraph?... [Digraphs]

Directed Graph... Binary relation, R on A: R ⊆ A x A with direc…

What is a path?... [Digraphs]

Sequence of vertices connected with edges from vertex A to ve…

Proof

Implication

Converse of Implication

Contrapositive of Implication

A sound argument in mathematics.

A Statement of the form: "If p then q" ... p is the hypothesis a…

Reverse of the hypothesis and conclusion ... EX: "If you are Am…

Formed by both reversing and denying the hypothesis and concl…

Proof

A sound argument in mathematics.

Implication

A Statement of the form: "If p then q" ... p is the hypothesis a…

Conjunction, Λ, and

Disjunction, V, or

Exclusive disjunction, ⊕, exclusive or

Conditional, →, if...then

p Λ q is True only when both are True

p V q is False only when both terms are False

p ⊕ q is True only when exactly one is True

p → q is F when p is True and q is False

Conjunction, Λ, and

p Λ q is True only when both are True

Disjunction, V, or

p V q is False only when both terms are False

universal statement

"all positive numbers are greater than…

"for all animals a, if a is a dog, the…

conditional statement

certain property is trye for all elements in a set. "for all"…

universal statement

conditional

if one thing is true then some other thing also has to be true

universal statement

certain property is trye for all elements in a set. "for all"…

"all positive numbers are greater than…

universal statement

Set

Power Set

intersection -- ∩

union -- U

a collection of objects (or elements) separated by commas.

the set of all subsets

what is in common in BOTH sets. AND

what exists in ALL sets without repetition. In one set OR the…

Set

a collection of objects (or elements) separated by commas.

Power Set

the set of all subsets

Irrational Numbers

Rational Numbers

Natural Numbers

Whole Numbers

do not terminate or repeat, cannot be written as a fraction

numbers that terminate or repeat, can be written as a fraction

Counting Numbers

0 and counting numbers

Irrational Numbers

do not terminate or repeat, cannot be written as a fraction

Rational Numbers

numbers that terminate or repeat, can be written as a fraction

Set

Important Sets to remember

Equal sets

Empty Set

An unordered collection of objects, called elements or member…

N = Set of natural numbers... Z = set of integers... Z+ = Set of po…

Two sets are equal, if and only if they have the same element…

Set with no elements, called null set.... Denoted by ∅. Also de…

Set

An unordered collection of objects, called elements or member…

Important Sets to remember

N = Set of natural numbers... Z = set of integers... Z+ = Set of po…

Graph

Directed Graph

Multiple Edges

Loop

A graph G = (V, E) consists of a nonempty set of vertices, V…

A directed graph (or digraph) is a graph in which each edge i…

We say a graph has multiple edges if two or more edges connec…

A loop is a vertex that connects a vertex to itself.

Graph

A graph G = (V, E) consists of a nonempty set of vertices, V…

Directed Graph

A directed graph (or digraph) is a graph in which each edge i…

Algorithms

Union

intersection

Difference

An algorithm is a precise set of instructions (steps) for per…

The union of sets A and B, denoted by A∪B, is the set that co…

The intersection of sets A and B , denoted by A ∩ B, is the s…

Set Operations Definition (Difference) The difference of sets…

Algorithms

An algorithm is a precise set of instructions (steps) for per…

Union

The union of sets A and B, denoted by A∪B, is the set that co…

addition principle

adjacency matrix

bipartite

brute force

when events a and b can occur in A + B ways

a matrix with rows and columns labeled by graph vertices with…

graph whose vertices can be divided into two distinct sets

from starting vertex list every possible circuit and find its…

addition principle

when events a and b can occur in A + B ways

adjacency matrix

a matrix with rows and columns labeled by graph vertices with…

universal quantifier

existential quantifier

De Morgan's laws for quantifiers

set

a formal expression used in asserting that a stated general p…

"There exists an element x in the domain such that P(x)"

Precedence:... ∃ and ∀ has higher precedence over the other logi…

is an unordered collection of objects, where each object can…

universal quantifier

a formal expression used in asserting that a stated general p…

existential quantifier

"There exists an element x in the domain such that P(x)"

Divisible: b|a

Odd

Prime

Composite

a is divisible by b if there is an integer c such that bc=a

a is odd provided there is integer x s.t. a=2x+1

p is prime if p>1 and only positive divisors of p are 1 and p

a is composite provided there is a b such that 1<b<a and b|a

Divisible: b|a

a is divisible by b if there is an integer c such that bc=a

Odd

a is odd provided there is integer x s.t. a=2x+1

Statement

Negation

Conjunction

Disjunction

A statement (proposition) is a sentence that is either true o…

A negation of a statement p is the statement "not p" or "it i…

The conjunction of two statements p and q is the statement "p…

The disjunction of two statements p and q is the statement "p…

Statement

A statement (proposition) is a sentence that is either true o…

Negation

A negation of a statement p is the statement "not p" or "it i…

Mortage

Interest

Principal

Commerical bank

A conveyance of a leim against propery that becomes void upon…

The percentage changed of a sum of mondy that is borrowing

The total amoumt of money borrowed

A bank that offers services to the general public and to comp…

Mortage

A conveyance of a leim against propery that becomes void upon…

Interest

The percentage changed of a sum of mondy that is borrowing

Alabama Paradox

Population Paradox

New State Paradox

Standard Divisor

An increase in the total number of seats to be apportioned ca…

An increase in a state's population can cause it to lose a se…

Adding a new state with its fair share of seats can affect th…

The Standard Divisor is the average number of people per seat…

Alabama Paradox

An increase in the total number of seats to be apportioned ca…

Population Paradox

An increase in a state's population can cause it to lose a se…

universal quantifier

existential quantifier

De Morgan's laws for quantifiers

set

a formal expression used in asserting that a stated general p…

"There exists an element x in the domain such that P(x)"

Precedence:... ∃ and ∀ has higher precedence over the other logi…

is an unordered collection of objects, where each object can…

universal quantifier

a formal expression used in asserting that a stated general p…

existential quantifier

"There exists an element x in the domain such that P(x)"

Arrows impossibility theorem

Borda count

Condorcet candidate

Condorcet criterion

Mathematically impossible for a democratic voting method to s…

Each place on ballot is assigned points

Candidate preferred by majority in head to head comparison

If there a condorcet candidate they should be the winner of t…

Arrows impossibility theorem

Mathematically impossible for a democratic voting method to s…

Borda count

Each place on ballot is assigned points

Even

Odd

Prime

Rational

n is even if n=2k, for some integer k.

n is odd if n=2k+1, for some integer k.

n is prime if n>1 and if n=rs then r=1 or s=1, for any intege…

n is rational if n=p/q where p,q are integers and q does not…

Even

n is even if n=2k, for some integer k.

Odd

n is odd if n=2k+1, for some integer k.

isolated vertex

loop

parallel

adjacent

vertex unconnected by edge to any other vertex

edge that connects vertex to itself

two edges that connect the same pair of vertices

vertices that are connected by an edge

isolated vertex

vertex unconnected by edge to any other vertex

loop

edge that connects vertex to itself

Kennth Arrow

Llyod Shapely

Hugo Steinhaus

Leonhard Euler

Best known/most versatile mathematical economists and awarded…

One of the giants of modern game theory and known for the Sha…

He is polish and made the discovery of the Ham sandwich theorem

One of the greatest mathematical geniuses

Kennth Arrow

Best known/most versatile mathematical economists and awarded…

Llyod Shapely

One of the giants of modern game theory and known for the Sha…

universal conditional statement

universal statement... "for all"

conditional statement ... "if then"

existential statement... "there is"

both universal and conditional statement

a certain property is true for all elements in a set

if one thing is true, some other thing has to be true

given a property that may or may not be true, there is at lea…

universal conditional statement

both universal and conditional statement

universal statement... "for all"

a certain property is true for all elements in a set

Rational Number

Countably Infinite

Discrete Mathematics

Stable Assignment

A real number, r, is called rational if it can be expressed a…

A set is called countably infinite if its elements can be pla…

A collection of mathematical topics that examine and use fini…

Given a collection of n men, m¹..., mⁿ, and n women, w¹..., w…

Rational Number

A real number, r, is called rational if it can be expressed a…

Countably Infinite

A set is called countably infinite if its elements can be pla…

1. Acute Angle

2. Adjacent Angles

3. Area

4. Circle

An angle with a measure of less than 90 degrees.

Angles that share a common side and a common vertex.

The space within a closed plane figure, measured in square un…

The set of all points in a plane that are equidistant from a…

1. Acute Angle

An angle with a measure of less than 90 degrees.

2. Adjacent Angles

Angles that share a common side and a common vertex.

Statement

Negation

Conjunction

Disjunction

A statement (proposition) is a sentence that is either true o…

A negation of a statement p is the statement "not p" or "it i…

The conjunction of two statements p and q is the statement "p…

The disjunction of two statements p and q is the statement "p…

Statement

A statement (proposition) is a sentence that is either true o…

Negation

A negation of a statement p is the statement "not p" or "it i…