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UNB/Year 4/Semester 2/CS2333/2024-01-24.md

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+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)