Lecture 1 Summary

Logistics

Definition: Set

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

Examples:

Definition: Function

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.

Case study: Sudoku

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).