2
Getting Started — Set Basics
2.1
Opening quote
•
Mathematics may be defned as the subject in which we never know what we are talking about,
nor whether what we are saying is true
– Bertrand Russell
2.2
Outline of Course
•
Foundations oF mathematics
some understanding oF the structure oF math
sets, logic, prooF techniques
•
counting and probability
relationship between these topics
•
Functions and relations
Focus on characteristics oF relations
how are certain relations similar/diﬀerent?
•
matrices
arrays oF values with operations on them
•
graphs
explain brie±y
•
trees
special category oF graphs
2.3
A Closer Look at Foundations
•
any branch oF mathematics is based on:
undefned terms, defnitions, assumptions, theorems
•
an example From plane Euclidean geometry
defnition oF a circle
based on terms: set, point, plane
ultimately, some terms must go undefned (see Russell)
•
assumptions
again From plane Euclidean geometry
parallel postulate (see Russell again)
•
theorems — concepts that can be
proven
using earlier items
what do we mean by a prooF?
based on logic
we will spend some time on both logic and prooF
2.4
Introduction to Sets
•
start with
set
and
element
as undefned terms
but we have an intuitive notion oF what we mean
•
set notation
{
and
}
use oF dots
•
set membership
use oF
∈
symbol
use oF
/
∈
sets containing sets
5
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the empty set,
∅
use of
{}
for
∅
diﬀerence between
∅
and
{∅}
2.5
Set Equality
•
sets are equal if they contain the same elements
•
order is immaterial
examples
•
repetition is ignored
examples
2.6
Useful Sets and Properties
•
some common sets
Z
+
,
Z
nonneg
,
Z
,
Q
,
R
fairly informal discussion of each
note occasional use of
N
for
{
1
,
2
,
3
,...
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 Fall '06
 Carter
 Set Theory, Sets, Bertrand Russell, Basic concepts in set theory, Venn diagram examples

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