Triangle Inequality for Altitudes

by mrichard, Jul 6, 2019, 3:50 PM

In basic geometry, almost everyone learns the Triangle Inequality, which states that the sum of any two sides in a triangle must be greater than the third side. Algebraically, if the sides of a triangle are $a,$ $b,$ and $c,$ then we have the set of inequalities
$\begin{align*} a + b &> c\\ b + c &> a\\ c + a & > b. \end{align*}$ The triangle inequality shows up all over the place. There are vector versions of it once you learn a little more math, and it gets generalized even more once you learn about metric spaces in topology.

But, I won't go too deep into that today. This is just a short post describing how the typical triangle inequality used in plane geometry also works for altitudes in a triangle.

An altitude of a triangle, (or a "height" in more lax language) is a segment that goes from one vertex in a triangle to the opposite side, and is perpendicular to that opposite side.
$[asy] draw((0,0)--(3,0)--(2,2)--cycle,black+1.2bp); draw((2,2)--(2,0),blue); draw(rightanglemark((3,0),(2,0),(2,2))); label("$A$", (2,2),N); label("$a$", (1.5,0),S); label("$B$", (0,0),W); label("$b$", (2.5,1),NE); label("$C$", (3,0),E); label("$c$", (1,1),NW); label("$h_a$", (2,1),W); label(scale(0.8)*"Altitude $h_a$ from vertex $A$ to side $a$.", (1.5,-0.7)); [/asy]$

The whole reason we care about an altitude is because it helps us find the area. The area of a triangle is half the product of a base and the height. The length of the altitude is the height, and the segment it is perpendicular to is the base. So, the area of the triangle above is $\frac12 a\cdot h_a.$

But this is true for any of the bases and their altitudes. So, of the altitude from vertex $B$ to side $b$ is $h_b,$ the area is also $\frac12 b\cdot h_b,$ and similarly we would have $\frac12 c\cdot h_c.$ Let's say the area is some number $R.$ Then, we can rearrange each of the area expressions to say

$\begin{align*} a &= \frac{2 R}{h_a}\\ b &= \frac{2 R}{h_b}\\ c &= \frac{2 R}{h_c}. \end{align*}$
Now, just substituting the expressions on the right into our original set of triangle inequalities (and removing the common factor of $2R$ from each one), we get

$\begin{align*} \frac{1}{h_a} + \frac{1}{h_b} & > \frac{1}{h_c}\\ \frac{1}{h_b} + \frac{1}{h_c} &> \frac{1}{h_a}\\ \frac{1}{h_c} + \frac{1}{h_a} &> \frac{1}{h_b}. \end{align*}$
And so the triangle inequality for altitudes is a harmonic version of the triangle inequality for the sides of a triangle. This is reminiscent of resistors in series versus those in parallel; in the first case, you just sum the resistances. In the second case, you sum their reciprocals to get the reciprocal of the total. It's fun to find these little symmetries in math.

Math Games and Puzzles

by mrichard, Mar 2, 2019, 3:53 AM

This is a collection of games and puzzles with very few prerequisites to get started. They can be adapted to fit many grade levels and knowledge, and be done in groups of many different sizes. I'll update it as more ideas come. Go to my personal website for a PDF of these activities.

Sprout

A game requiring 2 or more people.

Supplies: A piece of paper, writing utensils for each player. They can, but don't need to be, different colors.

Rules

Start by placing some dots on the paper. It doesn't really matter where or how many, but 4 to 8 is a good way to start.
On each turn, a player draws a line from one dot to another. Then, they place a new dot somewhere in the middle of their line.
Lines cannot cross.
Each dot can have at most three lines coming out from it. Once a dot has three lines emerging from it, cross it out and do not use it.
The last player to draw a line wins.

Sample Game:

We'll show a few steps of a game between Black and Orange, using 2 dots.
$[asy] unitsize(1cm); dot((0,0)); dot((1,0)); [/asy]$
Black will go first. They draw a line, and place a new dot on their line. Notice that Black's new dot counts as having two lines coming from it.
$[asy] unitsize(1cm); dot((0,0)); dot((1,0)); draw((0,0)--(1,0)); dot((1/2,0)); [/asy]$
Then, Orange will go.
$[asy] unitsize(1cm); dot((0,0)); dot((1,0)); draw((0,0)--(1,0)); dot((1/2,0)); draw((0,0)--(1/2,1)--(1,0),orange); dot((1/2,1)); [/asy]$
Then Black goes. They connect a dot that includes the dot Orange just made, so that dot is no longer available! Also, the right starting dot is unavailable. We make them large and red to make this clear. We continue in this way.
$[asy] unitsize(1cm); dot((0,0)); dot((1,0),red+5bp); draw((0,0)--(1,0)); dot((1/2,0)); draw((0,0)--(1/2,1)--(1,0),orange); dot((1/2,1),red+5bp); draw((1/2,1)--(1,1/2)--(1,0)); dot((1,1/2)); [/asy]$

Problem: Given the current arrangement of the game, Orange goes next. Can Orange guarantee a win for themselves at all? How will this game end?

Solution

Presentation Ideas:

Change the rules! What happens if you allow lines to cross, and increase how many lines can go through a dot?
Present these as puzzles. Create some nearly-completed games, and present the challenge to find the available winning move.
How does you strategy change when there are more than 2 people playing?

See-It-Say-It

This is a puzzle where students try to determine the rule for the following sequence:
$\begin{align*} &1\\ &11\\ &21\\ &1211\\ &111221\\ &312211\\ &\vdots \end{align*}$
The Pattern: The sequence is called ``See-It-Say-It" because that's how it is constructed.
We start with 1. The first term has ``one $1$ ", so we write down $11$ for the second term.
The second term has ``two $1\text{'s}$ ", so we write down $21$ for the third term.
The third term has ``one $2,$ one $1$ ", so we write down $12\ 11\implies1211$ for the fourth term. (Spaces represent a comma in the phrase).
The fourth term has ``one $1,$ one $2,$ and two $1\text{'s}$ ", so we write $11\ 12\ 21\implies 111221$ for the fifth term. And so on.

Presentation Ideas:

Typically refrain from telling them what the sequence is called. Start by putting the first three numbers down, and ask for the next term. Nearly every student will put $31.$
Surprise them by writing $1211$ instead. (I don't use commas because that makes spotting the pattern even more difficult.) Ask them again to work on finding the next term.
If they find the rule, or you decide to tell them, have them try to spot patterns or come up with their own variations.
Challenge them to find a term with a digit greater than $3$ in it (it will never happen.)

Magic Squares

A Magic Square is a square grid of numbers where every row, column, and diagonal sums to the same number. A true magic square will have all unique numbers. Here's the traditional $3\times 3$ magic square using the numbers 1 to 9, where the sum everywhere is 15:
$[asy] unitsize(1cm); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,black+2bp); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((1,0)--(1,3)); draw((2,0)--(2,3)); label("$4$", (0.5,2.5)); label("$9$", (1.5,2.5)); label("$2$", (2.5,2.5)); label("$3$", (0.5,1.5)); label("$5$", (1.5,1.5)); label("$7$", (2.5,1.5)); label("$8$", (0.5,0.5)); label("$1$", (1.5,0.5)); label("$6$", (2.5,0.5)); [/asy]$

Presentation Ideas: The first thing to do is challenge students to get the magic square above (of course it can be rotated/reflected). Give them the digits they are allowed to use; to make it a bit easier, also give them the target sum of 15.

An extra option is to "challenge" them to find a $2\times 2$ magic square where every number is different. It's impossible for pretty trivial reasons (if you fix the top-left corner, the squares to its right and below it must be equal), but it's good for them to come to that conclusion themselves.

Some other magic squares:

$3\times 3$ using 7, 8, 9, 12, 13, 14, 17, 18, 19. (Sum is 39.)
$4\times 4$ using 1 through 16. (Sum is 34.)
$3\times 3$ using 11, 17, 19, 23, 25, 27, 31, 33, 39. (Sum is 75.)

Also, you can challenge them to find a $3\times 3$ magic square where each entry is a unique perfect square. There is no known solution to that puzzle.

Three House Puzzle

Disclaimer: This puzzle is impossible with the assumed rules of using a single side of a sheet of paper.} But more on that later.

Supplies: Paper and pencil.

Rules: There are three houses. They each need gas, electricity, and water. However, these ``pipes" cannot cross. Can you connect each of the three utilities to each house without any lines crossing?

The typical diagram is given below.
$[asy] unitsize(1cm); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(0.5,1.5)--(1,1)); label("\Large$1$", (0.5,0.5)); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((2,1)--(2.5,1.5)--(3,1)); label("\Large$2$", (2.5,0.5)); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((4,1)--(4.5,1.5)--(5,1)); label("\Large$3$", (4.5,0.5)); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); label("\Large G",(0.5,2.5)); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); label("\Large E", (2.5,2.5)); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); label("\Large W", (4.5,2.5)); [/asy]$
Presentation Ideas:

Of course, you can just let them sit there frustrated. If they claim to have a solution on their single side of paper, you know there's an issue.
Loosen the restrictions. Make up stuff about "wormholes" (or whatever the alternate reality was in Stranger Things) to explain they can now try using the back of the paper! Suddenly it's possible.
What happens if you cut a hole in the middle of the paper to make it a donut?
Depending on what level of math they can understand, you can through words out there like "torus". Otherwise, you can let them know it's possible to do on a coffee mug.

Connect Four

Surprisingly easy to turn into a paper-and-pencil game using graph paper. If nothing else, it's a good strategy game to keep kids mentally engaged.

Supplies: Paper and pencil. Colors are fine, or students can play as $\times$ and $\circ.$

Rules: The standard game board is 6 holes deep by 7 holes wide. Students take turns ``dropping" pieces (they just need to imagine gravity exists on their paper) into the board. The first to make four in a row wins.

Presentation Ideas:

After playing for a little bit and developing a strategy, change the size of the game board.
Grab another color or make up another symbol, and try playing with more than 2 people. How does that change how you play? Everybody has 2 other people to try and defend against.
Remove ``gravity" from the game, and turn it into a larger version of tic-tac-toe. How does that work?
Other possible board variations include putting a ``free space" somewhere that either player can use; remove one (or two, or three) spaces in random places; with or without gravity, change the shape of the game board, making a rhombus or triangle or something to that effect.

Bulls and Cows

If you've played the board game Mastermind, this is the same thing.

Supplies: Paper and pencil. It's better for effect if you have some scraps of paper to write the target number down on.

Rules:

The short version: One player (P1) picks a (3, 4, 5, 6 digit number), the other player (P2) must try to guess it. I typically play without allowing 0 as a digit.
Logistically, P1 should write it down somewhere hidden where they can reference it. Then, P2 should write down each guess on another sheet of paper.
On each guess, P1 gives two pieces of information: How many correct digits are in the correct place, and how many correct digits are in the wrong place.
Visually, this is written down using $\bullet$ and $\circ.$ The former is correct digits in correct places, the latter is correct digits in the wrong place.
Traditionally, P1 chooses a 4-digit number, and P2 has 10 guesses. If they don't get it after 10 guesses, then P1 wins. It's proven every 4-digit number can be guessed with 7 chances, so you can make reduce the number of guesses to make it more stressful.
For other lengths of numbers, I'm unsure how many turns are necessary, but you could simply play two rounds and the player who guesses a number in fewer turns wins.

Let's suppose P1 chooses 1834, and P2 guesses 4823. Then P2 has a 4 and 3 in the incorrect spot, and an 8 in the correct spot. P1 would convey this information as shown:
$4823 \mid \bullet \circ \circ$ and P2 would then revise their guess in whatever way they see fit. The order in which P1 writes $\bullet$ and $\circ$ is immaterial.

Presentation Ideas:

You can just play the game!
Try to come up with a good strategy.
Come up with puzzles, where you have a string of guesses from which it is possible to find the chosen number. See below for a couple of examples.

Sample Puzzles

1. Try to find the number I chose, given the information below.

$\begin{tabular}{r|l} 137 & $\bullet$\\ 196 & $\bullet$ $\circ$\\ 165 & $\circ$ \end{tabular}$
Solution.

2. Find two possibilities for the number I chose, given the information below.

$\begin{tabular}{r|l} 1368 & $\circ$\\ 2154 & $\bullet$ $\bullet$ $\circ$\\ 2645 & $\bullet$ $\bullet$\\ 7145 & $\bullet$ $\bullet$ $\circ$ \end{tabular}$
Solution.

21 and Over

The only game here that's fully my invention. I made it for teaching mixed number addition in 4th grade math. It's probably a bit more fun when you actually know mixed numbers, but reports are fairly fine.

Supplies: Pencil and paper, and two standard dice (or sufficient number generator.)

Rules:

Roll two dice and add them together. This is the denominator of every mixed number you make.
Each player rolls one die. The person who rolls the biggest number goes first. Each player starts with a total of 0.
On your turn:
- Roll both dice. Choose one die to be the whole part of your mixed number and one die to be the numerator of your mixed number.
- Write down this mixed number and add it to your total.
- Let the other players check your addition, and fix it if you made an error.
If you are close to 21, you can choose to stop. The other players can still try to get closer to 21.
If you go over 21, you are out of the game.
The player closest to 21 who does not go over 21 wins!

Sample Game: Grogg and Winnie play a game of 21 and Over. When they roll the dice to start, they roll a 5 and 1. So, the denominator of every mixed number is 6.

Grogg rolls a 3 and Winnie rolls a 5, so Winnie goes first.

Winnie rolls a 3 and a 4. She makes the mixed number $4\frac{3}{6}.$ She writes it in a table like the one below. Her total is $4\frac{3}{6}.$

Grogg rolls a 5 and a 6. He makes the mixed number $5\frac{6}{6}.$ He writes it in the table as $6.$ His total is 6.

$\begin{tabular}{|c|c||c|c|} \hline Winnie's Numbers & Winnie's Total & Grogg's Numbers & Grogg's Total\\ \hline $4\frac{3}{6}$ & $4\frac{3}{6}$ & 6 & 6\\ \hline \end{tabular}$
The next few rounds are shown in the table.

$\begin{tabular}{|c|c||c|c|} \hline Winnie's Numbers & Winnie's Total & Grogg's Numbers & Grogg's Total\\ \hline $4\frac{3}{6}$ & $4\frac{3}{6}$ & 6 & 6\\ $5\frac{2}{6}$ & $9\frac{5}{6}$ & $5\frac{5}{6}$ & $11\frac{5}{6}$\\ $4\frac{1}{6}$ & $14$ & $2\frac{1}{3}$ & $14\frac{1}{6}$\\ $3\frac{2}{6}$ & $17\frac{2}{6}$ & $4\frac{1}{6}$ & $18\frac{2}{6}$\\ $2\frac{4}{6}$ & $20$ & &\\ \hline \end{tabular}$
Winnie decides to stop at 20.

Grogg is at $18\frac{2}{6},$ so he rolls again. He gets a 2 and a 4. He picks the number $2\frac{4}{6}.$ His new total is $18\frac{2}{6} + 2\frac{4}{6} = 21.$ Since Grogg got closer to 21 without going over 21, Grogg wins.

Presentation Ideas: Nothing immediately comes to mind, but you can always increase the number you're playing to. The first iteration went to 50, until it was decided that probably wasn't the best for fourth graders.

Factors Secret Mission

This is another puzzle I devised for fourth graders. It probably stands on its own fairly well for your casual person as well. It gets reasonably subtle at the end. Good luck.

The Puzzle: Find the value of $SECRET$ using the clues below.

$C,$ $E,$ $R,$ $S,$ and $T$ are digits from $0$ to $9$ . You can find them with the following clues:

$S$ is divisible by $7.$
$SE$ is divisible by $6.$
$SEC$ is divisible by $5$ and $2.$
$SECR$ is divisible by $4.$
$SECRE$ is divisible by $3.$
$SECRET$ is divisible by $2.$
$SECRETS$ is divisible by $9.$

Solution.

This post has been edited 1 time. Last edited by mrichard, Mar 2, 2019, 3:55 AM

Functions as Sets

by mrichard, Oct 30, 2018, 10:53 PM

In this little post, we'll expand on what we've learned about sets and functions. Specifically, we'll double-down on the claim that sets are vital to everything we do in mathematics. Functions are not just a way to desribe interactions between sets: functions are sets!

Many people will be familiar with the idea of an ordered pair. This is often how we describe "points" created by a function. When we write the ordered pair $(x,y)$ in the context of an existing function $f$ , we typically mean $f(x)=y.$ In this way, a function can be completely diescribed by its set of ordered pairs: $f = \{(x,y) \mid f(x) = y\}.$

This "set-builder" notation is new. Let's describe it briefly.

Now, we have a way to describe a function as a set. Namely, a function is just a set where every element is an ordered pair; each ordered pair describes a rule of the form $f(x)=y.$

This is a great first step! We are back to sets being the backbone of everything. Yet, there is something missing. All we've talked about are sets, and all of a sudden we introduce this "ordered pair." What are the rules about an ordered pair? What is it? As far as we can tell, it's some arbitrary object we created, unrelated to sets. As you may recall, a set is unordered. In set notation we write $\{x,y\} = \{y,x\}.$ So, how do ordered pairs relate to sets? Do they?

This will be our first dive into greater abstraction, analyzing an object we are intuitively familiar with in the context of another object we know as a foundation. In particular, we want to move from intuition to definition. To start, what is the defining aspect of an ordered pair?

As the name suggests, it is a pair of objects that happens to be in a certain order. Specifically, if we have two ordered pairs $(x_1,y_1)$ and $(x_2,y_2)$ , what does it mean to have $(x_1,y_1) = (x_2,y_2)?$ Intuitively, we would say "each of the parts of the pair are equal to the part in the othe pair." As a definition though, we would say $(x_1, y_1) = (x_2,y_2)$ means $x_1=x_2$ and $y_1=y_2.$

This is our defining rule for an ordered pair. Next, we need to figure out how to best describe this relationship using sets.

Here is the accepted mathematical definition.

This strikes most people as quite strange at first. But as we consider the implications of this definition, it eventually gives us an idea of "order" we want from an ordered pair. The first element is $x$ , the one that exists in both sets. The second element is the element that remains.

The payoff is we now have a full set-based definition of a function. We saw how a function is a set of ordered pairs, and we now see that an ordered pair is just a set as well. This set just happens to contain two more sets!

Here is our fully set-based definition of a function.

When doing mathematics, most people do not concern themselves with this definition in their daily work. It is good to see, to understand the foundation of what we do, but we come up with alternate notation like $(x,y)$ to make things a bit easier to communicate.

We can go even deeper, writing everything in terms of various logical statements. All of math can theoretically be written down using a symbol soup. But that is ultimately unhelpful for communication, and having a clearly communicated idea is one of the beauties of well-formulated mathematics.

Introduction to Functions

by mrichard, Oct 30, 2018, 4:51 AM

Previously, we talked about sets. Once we have these basic objects, we can start to create, define, and communicate relationships and patterns between sets. Mathematics uses the language of a function to describe these relationships.

Most simply, a function is a rule that takes elements on one set (often called inputs) and assigns them to elements of another set (often called outputs.) The one rule to remember to make sure you have a proper function is a function only has a single output for each input.

Let's look at a simple example of a function.

There are many notations used to discuss functions. The most common, and the one used throughout this blog, will be the form $f: A\to B$ . We call $f$ the function, where it takes elements from the set $A$ and assigns them to elements in the set $B$ .

So, we could call the function from our example $g: S\to F.$ While the letter used to name the function can be arbitrary, it is common to use $f$ , $g$ , or $h$ if no other information is given.

It is good to have a common language to describe the sets $A$ and $B$ in the context of a function.

Definition

Let's consider the function $g:S\to F$ from before. We say that $S$ is the domain, and $F$ is the codomain.

Imagine you wanted to tell a friend that our function $g$ took the input $2$ and assigned it to strawberries; however, you don't want to copy down the list of rules we had each time! This may not seem like a big deal, but imagine you had a domain with $1{,}000$ elements. We'd need an easier notation for these situations!

The accepted notation to tell our friends that $g$ sends $2$ to $\text{strawberries}$ is
$g(2) = \text{strawberries}.$ This is most commonly read as " $g$ of $2$ equals $\text{strawberries}.$ "

This notation is very flexible, as it allows us to have more than one element as the input. For example, since the subset $\{1,3\}\subset S$ has elements assigned to $\{\text{bananas},\text{apples}\} \subset F,$ we can say
$g(\{1,3\}) = \{\text{bananas},\text{apples}\}.$ Remember that a set does not have its elements in a specific order, so we can also write
$g(\{1,3\}) = \{\text{apples},\text{bananas}\}.$ Given just the above, we cannot tell exactly what $f(1)$ and $f(3)$ are; if we need to communicate that information, we can list them. But there are many situations where you just need to know if a particular element (very frequently $0$ or $1$ ) is among the outputs of a set of inputs.

Definition

As noted before, given the function $f:A\to B$ , there is no guarantee that every element in $B$ is actually an output of the function! We are only guaranteed that $B$ contains every output. So, it is then natural to wonder what exactly $f(A)$ is. Since $A$ is the domain (i.e. the set of all possible inputs) of $f$ , every element in $A$ is assigned to some element in $B$ . So, $f(A)$ must be a subset of $B$ , and in particular is the set that contains all possible outputs of the functions, and no extra elements.

Definition

Since all possible outputs must exist in the codomain $B$ , for any function $f:A\to B$ we must have $f(A)\subset B.$

Here is another example.

It is also good to note that a function can assign different inputs to the same output!

Here is a function where the codomain only has one element.

Questions

Consider the function $g:\{1,2\}\to \{1,2,3\}$ defined by $g(1)=3$ and $g(2)=1.$

What is the cardinality of this function's domain?

What is the codomain of the function?

What is the image of the function?

Consider another function $h:\{1,2\} \to \{1,2,3\}$ .

How many ways can we define this function so that every input is sent to a different output?

Consider a function $\iota: \{1\} \to A.$ Suppose $|A| = n.$

How many ways can we define this function?

Introduction to Sets

by mrichard, Oct 30, 2018, 4:08 AM

Preface to this Blog

Quite a while ago, an endeavoring individual tried to start an open-source repository of mathematical information called Mathbook. I contributed an article, which I'll put down in two parts on this blog as a start.

It seems that the project has died, although the website is still available. While this is a bit of a shame, I would like to give some of my own little lessons here. The creator's idea behind Mathbook was to focus on giving people an understanding of why we do math in a certain way. This is missing from mathematical curriculum today, but it is vital to understand that when math was developed, decisions were made for specific reasons.

In particular, while AoPS regulars probably don't concern themselves with these questions (they're pretty intrinsically motivated), the general populace and self-dubbed "Non Math People" do not have the same appreciation for mathematics. Efforts to spice up curriculum with applied questions always leads to contrived problems that, to me, really miss the entire point of mathematics.

Yes, mathematics is a very deductive, rational field. But it is also incredibly creative and inventive. Math is its own reward, its own justification (as discussed by Hardy in his Mathematician's Apology.) I want people to see some of the cool stuff that happens in math, but also build stuff up from their basics. I'll get a lot wrong, miss some explanations here and there, but I hope people will come along for the ride every once in the while. This will be a long shot from the Infinite Napkin, partially because I don't have the encyclopedic knowledge of Evan, but it will hopefully fill a small gap I see in how mathematics is viewed on a wider scale, that is generally not addressed by AoPS.

Introduction to Sets

We'll learn the basics of how sets are used in mathematics. It is important to understand basic arithmetic before diving in, but nothing else really. Just an openness of thought.

Understanding sets

In any field of mathematics, it is important to be able to deal with objects and structures. At the lowest level of mathematical objects and structures are sets. (Of course, one could dive into categories as a more general "basic" object, but we'll stick with the sets.)

Most simply, a set is a collection of objects. We can think of the set of all flowers in Hawaii, or the set of whole numbers between 10 and 37. Typically, we use curly brackets (braces) to denote a set, such as $\{1,2,3\}.$ If we are using the same set many times in a row, or talking about a set that cannot easily be written down, we can use some other symbol. Throughout this post we will let $S$ be the set $\{1,2,3\},$ and $H$ be the set of all flowers in Hawaii.

There are certain rules and terms used with sets that allow mathematicians to be consistent when using and talking about sets. For example, we want to know what to call the objects in our sets, and how we can write such a relationship down.

Definition

It is also natural to discuss how many elements are in a set.

Definition

For this tutorial, we will only be looking at sets with finite cardinality; this means we will always be able to list and count every element in the set. Future posts will explore larger sets, which becomes an even more powerful (and fun!) mathematical tool.

Often we want to look at some of the elements in a set, but not all of them. For example, we might want the elements of $H$ which are red flowers. This is a very common notion in mathematics: given an object or structure, how can we look at smaller objects that have a similar structure?

Definition

When doing mathematics, it is good practice to look at the simplest example of any object you are interested in exploring. When it comes to sets, it becomes natural to ask "What if my set has no elements?"

Definition

Almost everything you see and do has sets hiding in the background. They are a universal way of communicating mathematical ideas and structures, and are thus very important to understand.

Questions to Ponder

Is a set a subset of itself?

Is the empty set a subset of any other set?

So You Have Math Anxiety?