1. Review of Mathematical Notation / Terminology
Sets, Venn Diagrams, Sequences, Tuples, Functions, Relations, Graphs, Strings, Languages, Boolean Logic

2. Sets Order doesn’t matter In a set, repeats are “not allowed”
{7, 6, 5} and {5, 6, 7} are the same.
In a set, repeats are “not allowed”
{7, 7} is really {7}, i.e., they describe the same set.
In a multiset, repeats are allowed
{7, 7} and {7} are different

3. Sets
Empty set notation?
Union
Intersection
Compliment
Set Difference?

4. Venn Diagrams
Starts with…
Ends with..
Contains…
Questions…

5. Sequences Like sets, but the order matters and repeats are “allowed”
(5, 4, 7) is a different sequence than (4, 5, 7), but they would be the same set.
(5, 5, 5, 6) is a different sequence than (5, 5, 6) but they are the same set.

6. Tuples Its just another way of describing sequences.
2-tuple is a pair
3-tuple is a trio
Question: If A = {1,2} and B= {x,y,z} what is A X B?
X is the Cartesian product.
Note: This will create a set of pairs, 2-tuples, or sequences of size 2.

7. Power Set A = {0, 1, 2} Power set of A is
{ {}, {0}, {1}, {2}, {0,1}, {1,2}, {2,0}, {0,1,2}}
“Power Sequence” of A is
{ (), (0), (1), (2), (0,1), (1,0), (1,2), (2,1)… (0,1,2), (1,2,0), (2,0,1), (2,1,0), …)
Question: What is the size of the set above?

8. Functions f(a) = b Also called a mapping Function: Domain  Range
Abs: Z  Z
Add: Z X Z  Z
Division: Z X Z  Rational Numbers
Question: Example 0.8, 0.9, and 0.10

9. Relation Function whose Range is {TRUE, FALSE} is called a Predicate
Predicate whose Domain is a tuple is called a Relation.
If the Domain is a 2-tuple or pair, then its called a Binary Relation
Example: Equality of two numbers
Java: a == b or a.equals(b)
f(a,b) = true if a equals b, otherwise false
aRb, where R is the equality Relation
F: Z X Z  {TRUE, FALSE}

10. Equivalence Relation Satisfies three conditions
Reflexive: xRx is always true
Symmetric: if xRy is true, then yRx is true
Transitive: if xRy and yRz are true, then xRz is true.
Problems: Are the following Relations equivalence relations:
Equality x == y
Less-than x < y
F(x,y) = true if x+y is even, otherwise false

11. Graphs Directed vs. undirected Nodes/vertices Edges Degree
Labeled graph
Sub-graph
Path
Cycle
Simple cycle
Tree
Root node
Leaf nodes
Strongly connected directed graphs

12. Languages Alphabet notation No quotes Empty string Substring
Concatenation
Lexiographic ordering

13. Boolean Logic
And Or
Not
XOR
Distributive law