Review of Mathematical Notation / Terminology

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

2. Sets • Order doesn’t matter • {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 isA 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