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Study sets matching "discrete math"

25 terms
Discrete Math
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…
37 terms
Discrete Math
p --> q... p... So q
p --> q... ~q... So ~p
x1=x x0=0
not(not x)
modus ponens
modus tollens
dominance
x
p --> q... p... So q
modus ponens
p --> q... ~q... So ~p
modus tollens
53 terms
DISCRETE MATH
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
53 terms
DISCRETE MATH
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
25 terms
Discrete Math
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…
11 terms
Discrete Math
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…
30 terms
Discrete Math
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…
35 terms
Discrete Math
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
6 terms
Discrete Math
set of integers
non-negative integers
rational numbers
set of real numbers
Z
N
Q
R
set of integers
Z
non-negative integers
N
8 terms
Discrete Math
population
sample frame
p; what you should get
<p; what you get in the experiment
everyone you want to know about
people chosen
population parameter
sample statistic
population
everyone you want to know about
sample frame
people chosen
27 terms
Discrete Math
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…
17 terms
Discrete Math
de morgan's law
¬(E ∪ F)
¬(E ∩ F)
Converse given p →q
¬E ∩ ¬F
¬E ∪ ¬F
q →p
de morgan's law
¬(E ∪ F)
¬E ∩ ¬F
27 terms
Discrete Math'
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;
21 terms
Discrete Math
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…
9 terms
Discrete Math
Modus Ponens
Modus Tollens
Disjuntive Syllogism
Chain Rule
p , p→q infers q
p→q, ~q infers ~p
p∨q , ~p infers q
p→q, q→r infers p→r
Modus Ponens
p , p→q infers q
Modus Tollens
p→q, ~q infers ~p
19 terms
Discrete Math
Variance Formula
E(x) in sig. notation
Indistinguishable items, distinguishab…
Distinguishable items, distinguishable…
E(x²)-E(x)²
∑x·P(x)
n+k-1 choose k-1
boxes ^ balls
Variance Formula
E(x²)-E(x)²
E(x) in sig. notation
∑x·P(x)
36 terms
Discrete Math Basic Logic
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…
8 terms
Discrete math
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
43 terms
discrete math
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
8 terms
Discrete Math
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
Discrete Math Vocabulary
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
17 terms
Discrete Math
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…
17 terms
Discrete Math
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…
12 terms
discrete math
area of a trapezoid
volume of a cylinder
Volume of a rectangular prism
volume of sphere
b1+b2/2xH
pie.r^2xh
lxwxh
4/3x3.14xr^3
area of a trapezoid
b1+b2/2xH
volume of a cylinder
pie.r^2xh
11 terms
Discrete math
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…
29 terms
Discrete Math
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…
16 terms
Discrete math
Graph
Vertices
Edges
Degree of a vertex
Consists of a finite set of points
Set of points
Line segments or curves
The number of edges at that vertex
Graph
Consists of a finite set of points
Vertices
Set of points
8 terms
discrete math
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)"
24 terms
Discrete Math
Walk
Trail
Path
Closed Walk
alternating vertex/edge set
walk with no repeated edges
trail with no repeated vertices
starts and ends on the same vertex
Walk
alternating vertex/edge set
Trail
walk with no repeated edges
46 terms
Discrete Math
vertices
edges
directed graph
degree
the points of the graph
the lines that connect the vertices
graph that has directed edges, and order matters
how many edges are going in or out of the vertex
vertices
the points of the graph
edges
the lines that connect the vertices
57 terms
Discrete Math
not p
p and q
p or q
p or q but not both p and q
∼p
p∧q
p∨q
p ⊕ q or p XOR q
not p
∼p
p and q
p∧q
72 terms
Discrete Math
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
82 terms
Discrete Math
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…
17 terms
Discrete Math
Dilation
Translation
Reflection across the x-axis
Reflection across the y-axis
enlarges or shrinks by factor c
shifts horizontally by a, vertically by b
(x, y) --> (x, -y)
(x, y) --> (-x, y)
Dilation
enlarges or shrinks by factor c
Translation
shifts horizontally by a, vertically by b
11 terms
Discrete math
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
discrete math
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…
8 terms
discrete math
Modus Tollens
Transitivity
Elimination
Conjunction
p -> q... ~ q... ∴ ~p
p -> q... q -> r... ∴ p -> r
p V q... ~p... ∴ q
p... q... ∴ p ^ q
Modus Tollens
p -> q... ~ q... ∴ ~p
Transitivity
p -> q... q -> r... ∴ p -> r
13 terms
discrete math
modus tollens (contrapositive)
modus ponens (law of detachment)
law of addition
simplification
p -->q and ~q... ...~p
p-->q and p... ...q
p... ...p or q
p and q... ...p... ...q
modus tollens (contrapositive)
p -->q and ~q... ...~p
modus ponens (law of detachment)
p-->q and p... ...q
8 terms
discrete math
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)"
13 terms
Discrete math
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
9 terms
Discrete Math
Tree
Forest
Leaf
Cut-point
connected graph without cycles
graph without cycles
a vertex with one degree
each vertex of degree more than one
Tree
connected graph without cycles
Forest
graph without cycles
8 terms
Discrete Math
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.
62 terms
Discrete Math
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
11 terms
Discrete Math
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…
24 terms
discrete math
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
30 terms
discrete math
Order of operations
Converse of p->q
Contrapositive of p->q
Inverse of p->q
1. ~... 2.^... 3.v
q->p
~q->~p
~p->~q
Order of operations
1. ~... 2.^... 3.v
Converse of p->q
q->p
40 terms
Discrete Math
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…
100 terms
Discrete Math
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.
82 terms
Discrete Math
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…
26 terms
Discrete Math
p ∧ T ≡ p ... p ∨ F ≡ p
p ∨ T ≡ T ... p ∧ F ≡ F
p ∨ p ≡ p ... p ∧ p ≡ p
¬(¬p) ≡ p
Identity laws
Domination laws
Idempotent laws
Double Negation laws
p ∧ T ≡ p ... p ∨ F ≡ p
Identity laws
p ∨ T ≡ T ... p ∧ F ≡ F
Domination laws
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