Chapter 2: Sets, Functions, Sequences, and Sums. §2.1: Sets. § 2.1 – Sets. Introduction to Set Theory. A set is a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.
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(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
2
0
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8
1
1
Even integers from 2 to 9
3
5
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9
Odd integers from 1 to 9
Primes <10
Positive integers less than 10
Integers from 1 to 9
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
Example:{1,2} {1,2,3}
S
T
Venn Diagram equivalent of ST
§ 2.1 – Sets
(c)20012003, Michael P. Frank
}
Very
Important!
§ 2.1 – Sets
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N
Z
R
§ 2.1 – Sets
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§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
René Descartes (15961650)
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.1 – Sets
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
2
5
3
7
Union ExamplesThink “The United States of America includes every person who worked in any U.S. state last year.” (This is how the IRS sees it...)
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
2
4
6
5
Intersection ExamplesThink “The intersection of University Ave. and W 13th St. is just that part of the road surface that lies on both streets.”
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
SetAB
Set A
Set B
Set Difference  Venn Diagram§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
A
U
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
To prove statements about sets, of the form E1 = E2 (where Es are set expressions), here are three useful techniques:
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
Example: Show A(BC)=(AB)(AC).
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
Prove (AB)B = AB.
È
È


È
È


A
B
A
B
(
A
B
)
B
A
B
A
B
A
B
(
A
B
)
B
A
B
0
0
0
1
1
0
1
1
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
Prove (AB)C = (AC)(BC).
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
For an enumerable u.d. U with ordering {x1, x2, …}, represent a finite set SU as the finite bit string B=b1b2…bn wherei: xiS (i<n bi=1).
E.g. U=N,S={2,3,5,7,11}, B=001101010001.
In this representation, the set operators“”, “”, “” are implemented directly by bitwise OR, AND, NOT!
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
Symmetric Difference of A and B, denoted as AB,where
AB={x  x in A or in B, but not both}.
E.g: AB=(AB)(A B)
=(AB) (BA)
Do it in homework!
§ 2.2 – Set Operations
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
B
f
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f
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a
b
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x
A
Bipartite Graph
B
Plot
Like Venn diagrams
Graphical Representations§ 2.3 – Functions
(c)20012003, Michael P. Frank
Another example: →((T,F)) = F.
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
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§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
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§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
May Be
Larger
§ 2.3 – Functions
(c)20012003, Michael P. Frank
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Not even a function!
Not onetoone
Onetoone
OnetoOne Illustration§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
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§ 2.3 – Functions
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Onto(but not 11)
Not Onto(or 11)
Both 11and onto
11 butnot onto
Illustration of Onto§ 2.3 – Functions
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§ 2.3 – Functions
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§ 2.3 – Functions
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Identity Function Illustrationsy
x
Domain and range
§ 2.3 – Functions
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§ 2.3 – Functions
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§ 2.3 – Functions
(c)20012003, Michael P. Frank
.
1.6=2
2
.
1.6
.
1
1.6=1
0
.
1.4= 1
1
.
1.4
.
2
1.4= 2
.
.
.
3
3
3=3= 3
Visualizing Floor & Ceiling§ 2.3 – Functions
(c)20012003, Michael P. Frank
Note that for f (x)=x, the graph of f includes the point (a, 0) for all values of a such that a0 and a<1, but not for a=1. We say that the set of points (a,0) that is in f does not include its limit or boundary point (a,1). Sets that do not include all of their limit points are called open sets. In a plot, we draw a limit point of a curve using an open dot (circle) if the limit point is not on the curve, and with a closed (solid) dot if it is on the curve.
§ 2.3 – Functions
(c)20012003, Michael P. Frank
Set of points (x, f(x))
+2
3
x
+3
2
Plots with floor/ceiling: Example§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.3 – Functions
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
5 (the 5th smallest number >0)
11 (the 6th smallest odd number >0)
13 (the 6th smallest prime number)
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
(Distributive law.)
(Applicationof commutativity.)
(Index shifting.)
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
(Series splitting.)
(Order reversal.)
(Grouping.)
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
LeonhardEuler(17071783)
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
n+1
…
n+1
n+1
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
Geometric series.
Euler’s trick.
Quadratic series.
Cubic series.
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
Georg Cantor
18451918
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
A postulated enumeration of the reals:r1 = 0.d1,1 d1,2 d1,3 d1,4 d1,5 d1,6 d1,7 d1,8…r2 = 0.d2,1 d2,2 d2,3 d2,4 d2,5 d2,6 d2,7 d2,8…r3 = 0.d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7 d3,8…r4 = 0.d4,1 d4,2 d4,3 d4,4 d4,5 d4,6 d4,7 d4,8…...
Now, consider a real number generated by takingall digits di,i that lie along the diagonal in this figureand replacing them with different digits.
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank
§ 2.4 – Sequences and Summations
(c)20012003, Michael P. Frank