Lecture 1: Introduction

What is 2800 about?

How to succeed in 2800

Sets

A set is a collection of elements. Any object is either in a set or not; and two sets are the same if exactly the same objects are in both of them.

This means that sets do not have order: 1 is in both \(\{1,2,3\}\) and \(\{3,2,1\}\), as are 2 and 3; nothing else is in either of them. Thus they are the same set.

This also means that sets ignore duplicates: 1 is in both \(\{1,2,3\}\) and \(\{1,1,1,2,3,1,2,1\}\), as are 2 and 3, and nothing else, so they are the same set.

Set notation

Functions

Definition: A function is an unambiguous rule; for every input there should be an unambiguous output. The domain is the set of inputs. The codomain describes the type of outputs; there may be elements of the codomain that do not have an element of the domain that map to them. The image is the set of outputs that actually have an input mapped to them.

We'll discuss functions in more detail next lecture.

Note: Our definition of function corresponds to MCS's definition of total function.

Note: some books use "range" to mean "codomain", while others use "range" to mean "image". I try to avoid the term "range" to avoid ambiguity.

Sudoku

Sudoku is a popular puzzle that involves filling in a grid of numbers while satisfying certain rules. In the next lecture we'll use the notions of sets and functions to model solved sudoku puzzles. In the mean time, try a sudoku puzzle if you haven't before, and think about how you would model a solved board.