McCombs Math 381
Basic Counting Principles
1.
The Product Rule: Suppose a procedure can be broken down into a finite
sequence of tasks, T1, T2 , T3 , ! ! !, Tm , where each task in the sequence can be
performed in a finite number of ways, regardless of ho
McCombs Math 381
Relations on Sets
Basic Idea:
Given a set A , a relation R defined on set A is a rule that describes how to
compare the elements of set A to one another. In some cases, this rule will be
described explicitly by some formula or equation. I
McCombs Math 381
The Pigeonhole Principle
The Pigeonhole Principle: Let A and B be finite sets with a function f : A ! B .
( i)
( ii)
Basic Idea:
If A > B , then f is not one-to-one.
If A < B , then f is not onto.
If n objects are put into k boxes, where
McCombs Math 381
More Counting Examples
1.
In a technician's box there are 400 VLSI chips, 12 of which are faulty. How many
ways are there to pick two chips, so that one is a working chip and the other is
faulty?
! 12 $
! 388 $
There are # & ways to pick
McCombs Math 381
Equivalence Relations and Partitions
Given a set A , and a relation R on set A , we say th at R is an
equiva lence relat ion provided R is reflexive, sy mmetric and transitiv e.
Forma l D efinit ion:
Important Vocabulary: Suppose
1.
R is
McCombs Math 381
Boxes
Distinguishable objects into distinguishable boxes.
The number of ways to place n distinguishable objects into k distinguishable boxes, such
that n1 objects go in the first box, n2 in the second, . and nk go in the kth box is given
McCombs Math 381
The Binomial Theorem and Pascal's Triangle
The Binomial Theorem:
( x + y)
Let x and y be variables, and let n be a nonnegative integer.
! n$
! n$
! n$
! n $ n '1 ! n$ n
= # & x n + # & x n '1 y + # & x n ' 2 y 2 + ( ( ( + #
xy + # & y
" 0
McCombs Math 381
Proofs and Set Operations
Given finite sets A and B .
A ! B = x | x " A or x " B
1.
7.
cfw_
A ! B = cfw_( x, y ) | x " A, y " B
A ! B = ( A " B ) # ( B " A)
A = cfw_ x | x ! A
9.
A! B = A + B " A# B
11.
A! B"C = A! B # A!C
3.
5.
12.
8.
c
McCombs Math 381
Modular Arithmetic
Basic Idea:
Given an integer n ! 2 , we want to classify the set of integers according to
the reminder obtained when dividing by n .
Special Notation:
(
)
(
)
!a, b " Z , if n | a ! b , we write a ! b mod n ,
and say th
McCombs Math 81
Working with gcd(a,b)
Important Theorems:
Theorem:
Given a, b ! Z , with a and b not both 0. If d = gcd (a, b ) , then there
exist x, y ! Z such that d = ax + by .
Theorem:
Given a, b ! Z , with a and b not both 0. If d = gcd (a, b ) , the
McCombs Math 381
Greatest Common Divisor
Given a, b Z , with a 0 or b 0 . The greatest common divisor of a
Formal Definition:
and b , written gcd ( a, b ) , is the largest integer that divides both a
and b . In other words, gcd ( a, b ) = d means that d |
McCombs Math 381
Working with Functions
Given sets A and B , with a relation f from A to B .
Important Definitions:
1.
f : A B is a function from A to B provided the following are both true:
a A, b B : f (a ) = b
(i)
(ii)
If f (a ) = b and f (a ) = c, the
McCombs Math 381
Function Composition
1.
Suppose g : A B and f : B C where A = cfw_1,2,3,4, B = cfw_a,b,c, C = cfw_2,7,10,
and f and g are defined by g = cfw_(1,b),(2,a),(3,a),(4,b) and f = cfw_(a,10),(b,7),(c,2).
Find f g.
Ans: cfw_(1,7),(2,10),(3,10),(4
McCombs Math 381
The Chinese Remainder Theorem
Basic Idea:
We want to devise a strategy for finding simultaneous solutions to a
system of linear congruences.
Chinese Remainder Theorem:
Given pairwise relatively prime positive integers n1 , n2 , n3 ., nk ,
McCombs Math 381
Predicates and Quantifiers
Important Vocabulary and Notation:
Predicate:
The part of a propositional statement that refers to a property that
the statement of the subject can have.
e.g.
P : is a perfect square.
Propositional Function:
e.g
McCombs Math 381
Methods of Proof
Direct Proof:
To prove p q :
Assume p is true, then show this leads to q being true.
Proof by Contrapositive:
To prove p q :
Assume q is false, then show this leads to p being false.
That is, show that q p .
Proof by Cont
McCombs Math 381
Mathematical Induction
Main Idea:
We will develop a technique for proving statements about the set
of positive integers.
Proof Template:
To prove a statement of the form,
Every positive integer n satisfies the propositional function P ( n
McCombs Math 381
Contrapositive and Contradiction
Proof by Contrapositive:
To prove if A, then B:
Assume notB, and then show notA.
Proof by Contradiction:
To prove if A, then B:
Assume A and notB, and then show you get a
contradiction.
Prove each of the f