MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 6
Probl
Elementary Logic
Review Propositions, Logical Connectives:
Conjunction, Disjunction, Implication, Equiv
alence. Truth Tables. Tautologies, Logical Equivalence, and Contradictions. The
Laws of Logic.
T
Set Theory
Set theory is about the membership relation ; x S means x is a member of the
set S. This is another informal definition.
The statement S is a set consists of this
sentence plus a verificati
The Division Algorithm
Recall that the universe is Z.
Definition: For a, b Z we say a divides b
and write a|b if there exists a q Z such
that
b = qa .
In this case we also say that b is a <integer>
mu
Review
Lemma: For A, B, C S
P[A B C] = P[A] + P[B] + P[C]
P[A B] P[A C] P[B C]
+ P[A B C] .
Example: You toss n times a coin whose
Head comes up with probability p.
a) Describe a suitable state space
Review
Theorem (Division Algorithm): If a, b Z
with a > 0 then there exist unique integers
q, r such that
b = qa + r
and
0r<a.
Definition: A nonvoid subset I of Z is an
Ideal if for all a, b, z Z
a) a
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Elementary Logic
Logic is about Statements, also called
Propositions. We shall use the following
understanding of what a statement is:
Informal Definition: A statement or
proposition is a pair (d, m)
Review
Given a 6= b in our universe Z, there are
two generators g and g of the ideal
(a, b) def
= cfw_xa + yb : x, y Z
of all linear combinations of a, b. Both
of them are common divisors of a, b with
Lecture 3
Review: Principle 0, Principle 1, and
Theorem: A set of size n has
n def
n!
=
k!(n k)!
k
subsets of size k. Arrangements, Binomial
Theorem, and the Multinomial Formula:
(a1 + a2 + + ar )n
MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 4
Probl
MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 3
Probl
MATH 437: Principles of Numerical Analysis
Prof. Wolfgang Bangerth / Blocker 507D / [email protected]
TA: Youli Mao
/ Blocker 505E / [email protected]
Answers for homework assignment 2
Probl
Depicting Relations
Review
Exercise: If is a partial order on a set A
then so is its transpose T . It is called
the Reverse Order.
Example: Let A def
= cfw_2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
and defin
Laws of Set Theory
Let A, B U
A=A
Law of double complement
AB =AB
de Morgans law
AB =AB
de Morgans law
AB =BA
Commutative Law
AB =BA
Commutative Law
A (B C) = (A B) C Associativity
A (B C) = (A B) C A
The Integers - Induction
The Universe shall be the set Z of integers. It contains the natural numbers
N = cfw_0, 1, 2, . . .. We assume that we know
all about addition and multiplication of
integers,
Lecture 2
Review: Principle 0, Principle 1, and
Theorem: A set of size n has
n def
n!
=
k
k!(n k)!
subsets of size k if k n, none if k > n.
Pascals Triangle.
Example: To a conference on Middle East
Theorem: Let f : A B be a function.
Then A1, A2 A and B1, B2 B
B1 B2
A1 A2
f 1(B1 B2)
f 1(B1 B2)
f (A1 A2)
f (A1 A2)
= f 1(B1) f 1(B2) ;
= f (A1) f (A2) ;
= f 1(B1) f 1(B2) ;
= f 1(B1) f 1(B2) ;
= f (