1.4 KiB
1.4 KiB
Lecture Topic:
Equivalence Classes
You can represent equivalence classes by writing each one as a set:
C_0 = \{..., -10, -5, 0, 5, 10, 15, ...\} \text{ or } \{5p | p \in \mathbb{Z}\}C_0 = \{..., -9, -4, 1, 6, 11, 16, ...\} \text{ or } \{5p + 1| p \in \mathbb{Z}\}\text{and so on...}Graphs
Definitions:
- For each vertex v, the degree of v is the number of edges incident on v
- A path is a sequence of vertices connected by edges
- A cycle is a path that starts and ends at the same vertex
- A simple path is a path that has no repeated vertices
- A connected graph contains a path between every pair of vertices
- A disconnected graph is the opposite (look at slides for real def)
Regular Languages
A class of languages that can be processed by simple computers called finite automata
Simple Example of a finite automaton
Vending machine/toll booth gate Requires 25 cents or more to be deposited (gives no change)
Takes as input, 5 cent, 10 cent, 25 cent coins Uses states to remember how much has been deposited so far
States:
- q_0 : 0 cents deposited (START STATE)
- q_5 : 5 cents
- q_10 : 10 cents
- q_15 : 15 cents
- q_20 : 20 cents
- q_25 : 25 cents (ACCEPT STATE)
We represent the input as a string of symbols.
- n = nickel
- d = dime
- q = quarter Examples:
- ddn
- ndd
- dnd
- q Not accepted examples:
- nd
- d
\epsilon
State Diagram
(not really possible without an image in obsidian)