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MATH 185 Discrete Math
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Gravity
Course at McNeese State University using "Discrete Mathematics: 7th Ed" By Richard Johnsonbaugh
Key Concepts:
Terms in this set (98)
set
a collection of objects (order is not taken into account)
element
a member of a set

character representing "such that"
Z
character representing "the set of all integers"
Q
character representing "the set of all rational numbers"
R
character representing "the set of all real numbers"
cardinality
X = the number of elements in X
empty set
the set with no elements
subset
a set which contains only elements of another set
union
a set containing all of the elements found in either of two sets
intersection
a set containing all of the elements found in both of two sets
disjoint
describes two sets with a null intersection
universal set
the set which contains the space in which we work with other sets
complement
the set containing all elements not in another set
ordered pair
a set of two elements (a, b) which is distinct from (b, a)
Cartesian product
a form of set multiplication, resulting in a set of ordered pairs
function
assigns to each member of a set X exactly one member of a set Y
domain
X in f : X > Y
codomain
Y in f: X > Y
range
the set of y in Y such that f(x) = y for some x in X
onetoone (injective)
a function f: X > Y in which no more than one x in X is assigned to any y in Y
onto (surjective)
a function f: X > Y in which, for some x in X, f(x) = y for every y in Y
bijective
a function that is both injective and surjective
inverse
the function f^1: Y > X in relation to the onetoone, onto function f: X > Y
sequence
a function in which the domain consists of a set of consecutive integers
index
number describing the location of a term in a sequence
increasing
describes a sequence in which s(n) < s(n+1) for all n
decreasing
describes a sequence in which s(n) > s(n+1) for all n
nonincreasing
describes a sequence in which s(n) >= s(n+1) for all n
nondecreasing
describes a sequence in which s(n) <= s(n+1) for all n
subsequence
a sequence retaining only certain terms from another sequence, while maintaining their order
string
a finite sequence of elements from a set
null string
the string with no elements (lambda)
length
the number of elements in a string x, denoted x
concatenation
the string consisting of one string followed by another
substring
obtained by selecting some or all consecutive elements of a string
relation
a set of ordered pairs connecting two sets
reflexive
describes a relation on a set X in which (x,x) in R for every x in X
symmetric
describes a relation on a set X in which, for every x,y in X, if (x,y) is in R, (y,x) is in R
antisymmetric
describes a relation on a set X in which, for every x,y in X, if (x,y) in R and (y,x) in R, x = y
transitive
describes a relation on a set X in which, for every x,y,z in X, if (x,y) in R and (y,z) in R, (x,z) in R
partial order
a relation which is reflexive, antisymmetric, and transitive
equivalence relations
relations which are reflexive, symmetric, and transitive
recursive
describes a function which invokes itself
permutation
an ordering of objects
combination
a selection of objects with no regards to order
C(2n,n)/(n+1)
Catalan numbers
n!
number of ordered selections of n objects with no repetitions
C(n,r)
number of unordered selections of r objects from a set of n objects with no repetitions
n!/(n1! ... nt!)
number of ordered selections of n objects with repetitions described by the set (n1 ... nt)
C(k+t1,t1)
number of unordered selections of k elements, repetitions allowed, from among t items
binomial theorem
gives a formula for the coefficients in the expansion of (a+b)^n
pigeonhole principle
for function f: X > Y, with X = n and Y = m, there are at least ceiling(n/m) values (y1 ... yk) such that f(y1) = .... = f(yk)
recurrence relation
defines a sequence by giving the nth value in terms of certain of its predecessors
graph
a set of vertices and edges such that each edge is associated with an unordered pair of vertices
vertex
a node in a graph
edge
a line relating vertices in a graph
digraph
a graph consisting of edges associated with an ordered pair of vertices
incident
describes a vertex and edge which are connected
adjacent
describes two vertices connected by an edge
parallel
describes two edges associated with the same pair of vertices
loop
an edge incident on a single vertex
simple graph
a graph containing neither loops nor parallel edges
path
a sequence of edges and vertices connecting two vertices
connected graph
a graph with a path between every pair of vertices
subgraph
a graph containing only edges and vertices of another graph
simple path
a path with no repeated vertices
cycle
a path of nonzero length from a vertex to itself with no repeating edges
simple cycle
a cycle with no repeating vertices except for the beginning and ending vertex
Euler cycle
a cycle that includes all of the edges and all of the vertices of a graph
degree
the number of edges incident on a vertex
Hamiltonian cycle
a cycle that contains every vertex in a graph exactly once except for the starting and ending vertex
Dijkstra's algorithm
an algorithm for finding the shortest path between two vertices in a weighted graph
adjacency matrix
a representation of a graph in which element (i,j) is the number of edges incident on i and j, or twice the number of loops from i to j if i = j
incidence matrix
a representation of a graph in which element(i,j) is a 1 if edge j is incident on vertex i, and 0 otherwise
isomorphism
a rearrangement of a graph which retains all of its properties
planar
describes a graph which can be drawn without its edges crossing
face
a region of a planar graph bounded by edges, with no internal vertices or edges
series reduction
consists of replacing two edges (v1,v) and (v,v2) and their vertex v with the edge (v1,v2)
homeomorphic
describes two graphs, one of which can be reduced to a isomorphism of the other
Euler's formula
for a connected planar graph with e edges, f faces, and v vertices: f = e  v + 2
tree
a graph in which there exists a unique, simple path between any two vertices
rooted tree
a tree in which a particular vertex is given significance
level
describes the length of the path of the root of a tree to a vertex
height
the maximum level of any vertex in a rooted tree
parent
the vertex directly above another vertex in a rooted tree
ancestor
any vertex in the path from a vertex to the root of the tree
child
the vertex directly below another vertex in a rooted tree
descendant
a vertex below another, not connected through the root
sibling
another vertex with the same parent vertex
terminal vertex
a vertex with no children
internal vertex
a vertex with children
spanning tree
a subgraph which is also a tree, containing every vertex of the original graph
breadthfirst search
a type of search in which all vertices on a given level are processed before moving on to the next level
depthfirst search
a type of search which proceeds to successive levels in a tree at the earliest possible opportunity
Prim's algorithm
Finds Minimum Spanning Tree by adding the edge which connects to the nearest vertex to the current tree one at a time providing the edge would not form a cycle
binary tree
a tree in which each node has at most two children.
binary search tree
The left subtree of a node contains only nodes with keys less than the node's key.
The right subtree of a node contains only nodes with keys greater than the node's key.
Both the left and right subtrees must also be binary search trees.
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