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

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

Sets Empty set notation? Union Intersection Compliment Set Difference?

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

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.

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.

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?

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

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}

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

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

Languages Alphabet notation No quotes Empty string Substring Concatenation Lexiographic ordering

Boolean Logic And Or Not XOR Distributive law