partitions of sets:
The notion of a partition is that of a dividing up of a set into subsets, nonoverlapping.
A set A is partitioned by a collection S of subsets of A if
A = U cfw_X in S
the collection S is pairwise disjoint (i.e., for X and Y in S,
Introduction to Statistics
Overview: Statistical Inference, Samples, Populations,
and the Role of Probability
Beginning in the 1980s and continuing into the 21st century, a great deal of attention has been focused on improvem
Boolean algebra vs. truth tables:
In many instances, Boolean algebra is a much more efficient form of analysis
than truth tables
example: exercise 2.91:
(P & Q) => R) E (~(P & Q) v R)
(P & ~R) => ~Q) E (~(P & ~R) v ~Q)
E (~P v ~Q v R)
E (~P v R v ~Q)
negating restricted quantifiers:
What is the exact logical meaning of domain-restricted quantifiers?
(there exists x in S) P(x)
(for all x) (x in S => P(x)
(for all x in S) P(x)
(there exists x) (x in S & P(x)
So let's negate the firs
natural language with negation and implication
There can be serious ambiguities in interpreting natural (ordinary) language in
For instance, stating that there is one of something:
This can mean there is exactly one, or it can mean
more on =>
There are lots of ways to express P => Q in words:
if P, then Q
Q if P
P is sufficient for Q
Q is necessary for P
P implies Q
P only if Q
There are even expressions that have no visible "implies"-type words that
definition of continuity:
Let f be a real-valued function of a real variable (i.e., f(x) is a real number for x a
What does it mean to say f is continuous?
The general notion is that
as x -> a, f(x) -> f(a)
But there are several problem
size of power sets
If a |A| = n, then |P(A)| = 2^n
Suppose we know this for any n; what about for n + 1?
If |A| = n + 1, then pick one element of A-call it *-and let
A' = A - cfw_*
Then |A'| = n, so by supposition we know |P(A')| = 2^n
This course is about reading and writing mathematics-particularly proofs.
Working in groups is one way to generate ideas for proofs and to bounce ideas
off one another.
So some of what we do will be working in groups-in and out of class.
How many edg
proving the guess works
We know the following:
For any n:
E(n+1) = 2E(n) + 2^n
This is a recursion formula for E
Call it Recurs(n).
E(1) = 1
We want to know if the following is true:
For any n:
E(n) = n2^(n-1) ?
Call this equation Guess(n).
For sets A and B, we have the union of A and B:
For sets A and B, we the intersection of A and B:
A n B = cfw_x | x in A and x in B
For sets A and B, we have the set difference of A and B:
A u B = cfw_x | x in A or x in B
A - B = c
rational numbers and decimal expansion
A rational number (defined as a quotient of two integers) always has a decimal
expansion that eventually repeats.
The proof lies in looking at how the decimal expansion is developed, by
If x = m/n,
Group 3 (Kaiyue Fang, Pin Huang, Nolan Michniewicz, Caus Vedran)
Dr. Bradley Currey
Principles of Mathematics
Apr 25, 2016
Symmetries and Permutations
A permutation of a set A is a one-to-one and onto function from A to A. If A = cfw_1,2