Author:
Amanda Hager
Subject:
Mathematics
Material Type:
Lesson
Level:
Academic Lower Division
Provider:
University of Texas at Austin
Tags:
  • D2S2
  • Mindset
  • License:
    Creative Commons Attribution
    Language:
    English
    Media Formats:
    Text/HTML, Video

    Education Standards

    2.4 One-to-one correspondences

    2.4 One-to-one correspondences

    Overview

    TCCNS CourseMATH 1332: Contemporary Mathematics
    UT Austin CourseM 302: Introduction to Mathematics

    Introduction

    Suggested Resources and Preparation

    Materials and Technology

    • For the instructor: Mathematical manipulatives like those used in preschool or kindergarten classrooms can be very useful here. For example, coins or marbles. 

    Prerequisite Assumptions

    None

    Overview and Student Objectives

    Lesson Length

    50 minutes

    Lesson Objectives

    Students will understand that:

    • Sets are collections of objects, and sets have a size, or cardinality.
    • One to one correspondences between sets can be expressed in a variety of ways, if they exist.
    • Two sets have the same cardinality if we can find a one-to-one correspondence between them.

    Students will be able to:

    • Depict one-to-one correspondences between sets using tables. Use of functions for sets of numbers is optional but not necessary.
    • Argue whether "messy" real-world sets have a one-to-one correspondence with each other by discussing how objects of those sets might be paired.

    One-to-one correspondences

    Counting

    Take a moment and think about how you would answer this question:

    How do you teach counting?

    You may have siblings, nieces/nephews, children that you have taught and you may have some experience to draw on. Or perhaps you've worked as a sitter, nanny, or daycare teacher. But it's not often that we need to stop and really list out the steps for doing something like this.

    For example, there is a big difference between knowing "counting words" and performing the actual act of counting.

    Video: Violet counts

    Notice that she knows the English words "one" through "twelve" perfectly fine. But there are some other, more subtle, things that she isn't getting yet. For example, perhaps you noticed that there are these other rules:

    • You have to point at a single object every time you say a word. 
    • You can't point at the objects more than once.
    • You can't leave objects out.
    • It doesn't matter what order you point in.

    It's actually pretty complicated! Compare to this later example with the same child:

    Video: Violet counts!

    One to one correspondences/cardinality

    Terms

    Two sets have a one-to-one correspondence if you can draw or describe a perfect matching of one element of one set to one element of the other set, with no leftovers on either side.

    Two sets have the same cardinality or the same size if there is a one-to-one correspondence between them.

    Video: One to one corresondences

    Examples

    The set of children in the top photograph and the set of guinea pigs in the bottom photo have a one-to-one correspondence with each other.

    Two children in a chair eating cereal

    Two guinea pigs eating carrots
    Right Photo by Bonnie Kittle on Unsplash

    In plain English, you might say "yes because there are two of each." If I were forcing you to explain why there were two of each you might hand-count "See? One, two children. One, two guinea pigs." But to understand what comes next, we need to see this in a particular way: we could correspond one kid with one guinea pig with no leftovers, no repeats. It's not important how it gets done as long as there are no leftovers, no repeats.

    Top pair of images, left child paired with right guinea pig and right child paired with left guinea pig. Bottom pair of images, left child paired with left guinea pig and right child paired with right guinea pig.

    Video: Examples of sets that have/do not have one to one correspondence

    A note on grammar and other tips

    • Two sets have a one-to-one correspondence, they can't be a correspondence. This is similar to the word "connection," as in "Two computers can have a connection, but they can't be a connection."
    • Every set has a cardinality. Two sets either have the same cardinality or different cardinalities. It's improper to say "two sets have cardinality." Think of cardinality as size: you wouldn't say "two sets have size."
    • Consider any two sets you like. It's black-and-white: either those sets have a one-to-one correspondence or they don't.
    • If two sets have a one-to-one correspondence, then they have more than one (unless the sets have one element in them).

    Quiz Questions

    QuestionAnswer
    12
    21, 6
    34

     

    Question 1

    When do two sets have a one-to-one correspondence?

    1. When you can make a table where all the objects in one set are used up.
    2. When the objects in the sets can be paired evenly - no repeats, no leftovers.
    3. When you can make a table with objects from both sets in it.
    4. When you can count both sets.

    Question 2

    Which of these sets has the same cardinality as the alphabet? Select all that apply.

    1. \(\{1, 2, 3, 4, 5, 6, 7, 8, 9 ,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26\}\)
    2. \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26\}\)
    3. \(\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26\}\)
    4. \(\{1, 2, 3, 4,\dots\}\)   (The set of all natural numbers)
    5. \(\{26\}\)
    6. \(\{0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50\}\)

    Question 3

    Suppose a stranger tells you that the license plate number on her car is 498HTJ.  If you had a giant list of all cars in the United States together with their license plate numbers, would you be able to precisely identify the stranger's vehicle? (Context note: there are 50 states in the United States, and each state is responsible for making its own system of registering cars and drivers).

    1. Yes because there can only be a few cars in the U.S. that have that number, so it's likely we'll figure it out.
    2. Yes because our list has every car on it, so it has to be there somewhere.
    3. No because she's probably lying.
    4. No because we don't know what state her car is registered in.  There could be two cars with different states but same plate numbers.

    Homework Questions

    Question 2.4.1

    Two sets have a one-to-one correspondence if you can draw or describe a perfect matching of one element of one set to one element of the other set, with no leftovers on either side.

    Question 2.4.2

    Probably not. You would have to have exactly one driver assigned to each and every bus. If there are any broken buses with no drivers, or multiple drivers assigned to one bus, then there is no one-to-one correspondence.

    A numbers-based approach is also possible. CapMetro in Austin states that there are currently 368 MetroBuses. However, it's not as easy to find out how many drivers there are. If students perform some kind of estimate or research to get a number that is not 368, this is also proof.

    Question 2.4.3

    Probably not. There are many examples of steering wheels not on working vehicles: junkyard cars, boats with helms, fake or play cars (like for children). Also there are lots of examples of working vehicles without steering wheels: planes, certain boats, motorcycles, bicycles.

    Question 2.4.4

    The statement "every student must live in a dorm room" does not imply a one-to-one correspondence. We also need to know if there are any empty rooms, if any students share a room, and if any students have multiple rooms (the third one is unlikely).

    Question 2.4.5

    No. "U.S. residents" has many definitions, but the most common one is that of the U.S. Census Bureau, which says you are a resident if you live here. There are many groups of U.S. residents who don't have SSNs: international students, newborn babies, Amish children and teenagers, some undocumented folks, etc.

    Less common, but just as valid, you could cite that victims of identity theft are reissued SSNs but they still own the old one so that is a person who legally has two numbers.

    Question 2.4.6

    No. The set of U.S. residents is the same as from Question 5, but the set of numbers has changed. The set of all allowable numbers is the set of all nine-digit numbers, and there are a billion of those (10 multiplied by itself nine times). U.S. Census Bureau says there are 330 million or so residents, so no way can there be a one-to-one correspondence.

    Question 2.4.7

    Examples include the left student touching each dot on the die once and saying a number word, the left student moving the game piece one square at a time and saying a number word, or the correspondence between the dots on the die and the spaces she should move on the board.

    Preschool board game

     

    Question 2.4.1

    Define a one-to-one correspondence between sets.

    Question 2.4.2

    Is there a one-to-one correspondence between the set of all Austin city buses and the set of all Austin city bus drivers? In your response, please define or give some details about the sets "all Austin city buses" and "all Austin city bus drivers."

    Question 2.4.3

    Is there a one-to-one correspondence between the set of working vehicles and the set of all steering wheels?

    Question 2.4.4

    You are accepted to a small college, and you learn that every student must live in a dorm room. Does this fact imply that there is a one-to-one correspondence between students at this school and dorm rooms?

    Question 2.4.5

    Is there a one-to-one correspondence between the set of U.S. residents and the set of Social Security numbers being used by people? (Social Security is the public retirement fund in the U.S., and participants are given a nine-digit number. All nine digits can be 0 through 9.)

    sample social security card, personal information redacted.
    A sample Social Security card

    Question 2.4.6

    Is there a one-to-one correspondence between U.S. residents and the set of all allowable Social Security numbers (whether or not they're currently being used)?

    Question 2.4.7

    Application! Imagine you are working with a class of four-year-olds in a preschool. Simple board games are often used to reinforce one-to-one correspondences. Watch the follow video from 2:21-2:33 and describe two examples of one-to-one correspondences that you see the children using.