- Intro
- Logistics
- Starting definitions (with Sudoku example)
- key terms: set, cartesian product, function, domain, codomain

- mostly see the syllabus
- first homework released Friday sometime after class
- nobody is on CMS yet; I'll let you know
- please no laptops or other devices in lecture
- Prof. Hopcroft is traveling, he will lecture on Monday. We'll alternate throughout the semester.

Definition: a **set** is an unordered collection of **elements**. Two sets are considered the same if they contain the same elements; the order and duplication of elements is ignored. We write *e* ∈ *S* to indicate that *e* is an element of the set *S* (similarly $e \notin S$ indicates that *e* is *not* an element of *S*).

(Note: ∈ ("is an element of", `\in`

in LaTeX) is different from *e* or *E* (the letter e) and *ε* (the greek letter epsilon, LaTeX `\epsilon`

).)

There are many ways to write down sets.

- You can enumerate the elements: {1, 2, 3, 4, 5}
- You can use ellipses if it's clear: {1, 2, . . . , 5}
- You can give a (clear!) english description: "the set of all even natural numbers"
- We'll discuss set comprehension notation next time.

You have written down a set if the reader can unambiguously determine whether a given element is or is not in the set.

Examples:

- {1, 2} = {2, 1} = {1, 2, 2, 1, 2} ≠ {1, 2, 3}
- {1, 2} ≠ {3, 4}
- We let
*N*= {1, 2, . . . 9} for the sudoku example - We defined
*C*as the set of all pairs of numbers from*N*:*C*= {(1, 1), (1, 2), …, (1, 9), (2, 1), (2, 2), …, (2, 9), …, (9, 1), …, (9, 9)}

This set is called the**cartesian product**of*N*with itself, and is written*N*×*N*.

A **function** *f* from a set *A* to a set *B* is an *unambiguous* rule, which gives an element of *B* for *each* element of *A*. *A* is called the **domain** of *f*, while *B* is called the **codomain** of *f*. We often write *f*: *A* → *B* to represent the statement "*f* is a function from *A* to *B*".

As with sets, there are many ways to write down functions. The key rule is that everyone agrees (unambiguously!) what the output is for a given input (the input is the element of the domain, the output is the element of the codomain).

The domain and codomain are different: every element of the domain must have a corresponding element of the codomain, but an element of the codomain may have any number (including 0) of elements of the domain that map to it.

We started modeling sudoku solutions. Here's where we are so far:

Let the *set of numbers* *N* = {1, 2, …, 9}. Let the set of *cells* *C* = *N* × *N*.

Definition: a **sudoku solution** is a function from *C* to *N* satisfying certain properties (properties to be enumerated later).