---
tags: mth225, dailyprep
---

# Daily Prep 3.4 -- MTH 225

## Overview

Now that we know a little about functions, we're going to learn about different kinds of functions based on how the functions maps elements from the domain to the codomain. The three types of functions are **injective**, **surjective**, and **bijective** and we'll be learning how to tell which properties different functions have and why this matters.

## Learning objectives

**Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught.

* State the definitions of injective, surjective, and bijective functions.
* Given a function between two *finite* sets, determine whether it is injective, surjective, or bijective.
* Find values of the floor, ceiling, factorial, mod (`%`), and div (`//`) functions by hand and using Python 3.

**Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class.

* Give examples of a function between two sets that is injective but not surjective, surjective but not injective, and neither injective nor surjective.
* Determine whether it is not possible to construct a function between two sets that has a given property (injective, surjective, bijective) and if so, explain why.
* Given a function between two *infinite* sets, determine whether it is injective, surjective, or bijective.

## Resources for learning

**Video:** Watch these from [the MTH 225 playlist](https://vimeo.com/showcase/8667148) (total running time 31:09):

<iframe src="https://player.vimeo.com/video/614452153?h=58ba81862d" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>

<iframe src="https://player.vimeo.com/video/614525515?h=065c033a89" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>

<iframe src="https://player.vimeo.com/video/614541763?h=7c14d8d686" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>

<iframe src="https://player.vimeo.com/video/614660250?h=03e81cce6b" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>

**Text:** Read in the text, [Section 0.4](http://discrete.openmathbooks.org/dmoi3/sec_intro-functions.html) starting at the subsection "Surjections, Injections, and Bijections" and continuing to the end.

You are free to search for and use other resources in addition to, or instead of the above, as long as you can work the exercises below.

## Exercises

Once you have watched the videos above, go to this form and complete all the non-optional items on it:

https://docs.google.com/forms/d/e/1FAIpQLSexaIXP_pT4M3JYt4rkO2VXVK72-c6NkrBrTjApFo2gO3AH9Q/viewform

## Submission and grading

**Submitting your work:** Your work is submitted when you submit the Google Form. You should receive an email receipt indicating that the work was submitted successfully.

**How this is graded:** The pre-class portion of the Daily Prep is graded either 0 points or 1 point, on the basis of completeness and effort. Wrong answers are not penalized. Earning a "1" requires that you:

- Turn the work in before its deadline;
- Leave no item blank or skipped, even accidentally; and
- Give a good-faith effort at a correct answer on every non-optional item.

More information can be found in the [Specifications for Satisfactory Work in MTH 225](/Cy6P0rGZQzuOM3NwZ3ZuMw) document.

When you arrive for the class meeting, you'll be put into a group of 2-3 to complete a quiz over this material, which will be graded on a 0/1 scale on the basis of correctness.