1/30
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 coll
Chapter 1
Introduction to Statistics
and Probability
1.1
Overview: Statistical Inference, Samples, Populations,
and the Role of Probability
Beginning in the 1980s and continuing into the 21st century,
2/20
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) =>
2/15
negating restricted quantifiers:
What is the exact logical meaning of domain-restricted quantifiers?
means
(there exists x in S) P(x)
means
(for all x) (x in S => P(x)
and
(for all x in S) P(x)
(
2/8
natural language with negation and implication
There can be serious ambiguities in interpreting natural (ordinary) language in
mathematical terms.
For instance, stating that there is one of someth
2/6
more on =>
interpretations
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
2/13
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
real number).
What does it mean to say f is continuous?
The general notion is tha
1/25
size of power sets
If a |A| = n, then |P(A)| = 2^n
proof:
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
1/14
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
1/16
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
1/23
set constructions:
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
2/4
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 decima
Huang 1
Group 3 (Kaiyue Fang, Pin Huang, Nolan Michniewicz, Caus Vedran)
Pin Huang
Dr. Bradley Currey
Principles of Mathematics
Apr 25, 2016
Symmetries and Permutations
A permutation of a set A is a o