l2 supplementary notes page 1 04-02-2010: recap l sum principle n applied to selection sort l...
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Page 1L2 Supplementary Notes
04-02-2010: Recap
Sum Principle Applied to selection sort
Product Principle Applied to matrix multiplication and the next item
Two element-subsets
Set concepts and notations Sets, mutually disjoint sets, size, union, partition Set does not allow duplicates
Page 2L2 Supplementary Notes
04-02-2010 Recap: Sum Principle
Page 3L2 Supplementary Notes
04-02-2010 Recap: Product Principle
Si and Sj are disjoint, |Si| = n
S = S1 U S2 U … U Sm
|S| = m |Si| = mn
Page 4L2 Supplementary Notes
04-02-2010 Today First 3 items on Page 2 of “ More Counting”
Page 5L2 Supplementary Notes
More counting, Page 4 (MC 4)
Page 6L2 Supplementary Notes
Use of Product Principle in Entry Code Example (MC 4)
Page 7L2 Supplementary Notes
MC 5-9
Page 8L2 Supplementary Notes
Suppl 4, MC10, 11
Page 9L2 Supplementary Notes
Discrete Function (MC 11, 12)
S = {1, 2, 3}: domain of function f
T={Sam, Mary, Sarah}: range of function f
Page 10L2 Supplementary Notes
Notes (MC 11)
For each element s of S, f gives one element of T, f(s)
In general NOT: For each element of T,…
There may be t of T, such that f(s) \= t for all s of S
Only a special kind of function has this property, onto
Page 11L2 Supplementary Notes
MC 14
Page 12L2 Supplementary Notes
Exercise on Functions (MC 15)
All functions from {1, 2} ->{a, b}
Page 13L2 Supplementary Notes
Counting Functions (MC 16)
Page 14L2 Supplementary Notes
Counting Functions (MC17)
Page 15L2 Supplementary Notes
Injection (MC 18)
f: {1, 2} {a, b}
Page 16L2 Supplementary Notes
Surjection (MC 18)
f: {1, 2} {a, b}
Page 17L2 Supplementary Notes
Examples (MC 19)
Page 18L2 Supplementary Notes
Bijection (MC 20)
Domain and range have same number of elements.
Page 19L2 Supplementary Notes
Permutation (MC 20)
Page 20L2 Supplementary Notes
Bijection Principle
Counting elements in S
May be difficult directly
Find another set T that is easy to count
Define a function f: S T
Prove that f is a bijection
Count T
Page 21L2 Supplementary Notes
Three Increasing Triples
Page 22L2 Supplementary Notes
Increasing Triples
Page 23L2 Supplementary Notes
3-element subsets/3-element permutations
Page 24L2 Supplementary Notes
K-th falling factorial
Page 25L2 Supplementary Notes
k-element subsets/k-elemen permutations