1. Discrete Mathematics CS 2610
February 24, part 2

2. Summations- Properties
Index Shifting

3. Summations- Properties
Series Splitting
Order Reversal
Grouping
Oops, f(2i)

4. Double Summations 4 3 å å
ij= i = 1 j = 1

5. Summations Theorem: If a and r are real numbers, r0, then Proof:
Case I : r=1
Case II : r  1

6. Some Useful Summation Formulae
AMG
Geometric series.
Li’l Gauss’ trick.
Quadratic series.
Cubic series.

7. Cardinality
Def.: The cardinality of a set is the number of elements in the set.
Def.: Let A and B be two sets.
A and B have the same cardinality iff there is a one-to-one correspondence (bijection) between A and B

8. f(6)=a, f(3)=b, f(5)=c, f(3)=d, f(1)=e
Cardinality
Example
On finite sets
Let A = { 6, 3, 5, 4, 1 }
Let B = { a, b, c, d, e }
There is at least a one-to-one correspondence between the sets
Observe |A| = |B|.
We can define a one-to-one and onto function f:AB, for example:
f(6)=a, f(3)=b, f(5)=c, f(3)=d, f(1)=e

9. Cardinality On Infinite sets
Z+ and E={xZ+ | Even(x)} have the same cardinality
There is a bijection f between the two sets
f: Z+  E
f(x) = 2x
1 ↔ ↔ ↔ 6 4 ↔ 8
5 ↔ ↔ ↔ ↔ 16
f is one-to-one, for all x,y in Z+, if xy then 2x2y
f is onto: let y  E,
There exists k  Z+ , y=2k, f-1(y)=k

10. Countable Sets and Uncountable Sets
Def.: The set A is countable if it is finite or if it has the same cardinality as the set of positive integers. Otherwise it is uncountable.
(aleph) denotes the cardinality of infinite countable sets
Examples:
Infinite Countable Sets: N, Z+, Z-, Z
Infinite Uncountable Sets: R, R+, R-