CMSC 250
Discrete Structures
Mathematical Induction
Mathematical Induction
Used to verify a property of a sequence
28 June 2007
Mathematical Induction
2
Inductive Proof
Let P(n) be a property that is defined for
integers n, and let a be a fixed integer.
S
CMSC 250
Discrete Structures
Counting
Counting Elements in a List
How many integers in the list from 1 to 10?
How many integers in the list from m to n?
(assuming m n)
nm+1
Can you prove this?
12 July 2007
Counting
2
Prove: # elements in list
Base case (
CMSC 250
Discrete Structures
Functions
Terminology
Domain: set which holds the values to
which we apply the function
Co-domain: set which holds the possible
values (results) of the function
Range: set of actual values received when
applying the function t
CMSC 250
Discrete Structures
Relations
Relations
The most basic relation is = (e.g. x = y)
Generally x R y TRUE or FALSE
R(x,y) is a more generic representation
R is a binary relation between elements of
some set A to some set B, where xA and
yB
19 July
CMSC 250
Discrete Structures
Graphs and Trees
Graphs
Vertices
Edges (endpoints)
24 July 2007
Graphs and Trees
2
Types of Graphs
Directed order
counts when
discussing edges
Undirected
(bidirectional)
Weighted each
edge has a value
associated with it
Unweig
CMSC 250
Discrete Structures
Set Theory
Sets
Definition of a Set:
NAME = cfw_list or description of elements
Examples
B = cfw_1,2,3
C = cfw_xZ+ | - 4 < x < 4
Axiom of Extension
A set of elements is completely defined by
elements, regardless of order an
CMSC 250
Discrete Structures
Number Theory
Exactly one car in the plant has color
H(a) := a has color
xCars
H(x)
aCars
a x ~ H(a)
H(a,b) := a has color b
xCars
yColors
H(x,y)
aCars, bColors
a x ~ H(a,b)
20 June 2007
Number Theory
2
At most one car i
CMSC 250
CMSC
Discrete Structures
Predicate Calculus
Ambiguity
13 June 2007
Predicate Calculus
2
Examples
Everyone loves logic.
There is a person only a mom could love.
Every sufficiently large odd number can be
written as the summation of three primes.
1