Suppose we were interested in forming sets of Major League Baseball players.

What kinds of players would we include if we wanted to form the set of “outstanding left-fielders”? The first thing we would need to do is figure out which players are outstanding left-fielders. Well, who qualifies as an outstanding left-fielder? Are they the players who can pitch with the most accuracy? If so, what is the accuracy cutoff?

If accuracy is not the metric, what about running speed? How fast a player may be able to run to catch a ball may be important. What about the ratio of caught balls to missed balls? How exactly do we decide who to include in the set of outstanding left-fielders?

The problem with a “set” such as “outstanding left-fielders” is that the term outstanding is vague. Notice that we enclosed the word set with quotation marks in the previous sentence. That’s because part of the definition of a set is to be well-defined. Since it is not exactly clear how to determine who an outstanding left-fielder is, we have no well-defined criterion. As such, the above description does not define a set.

Let’s do something different. This time, let’s niche down into something like this:

Here, all we have to do is check the records of the players from the 1990s, which we can do here, and see who hit at least 300 home runs. Here are those players:

Notice the difference, one description was vague and open to interpretation, and as such, not everyone may agree on what players would be included. The other description was specific, and could be verified against the available records. The second description was not open to interpretation at all: we offered a very specific metric, and only those players meeting that requirement are included.

Let’s revisit the set defined in Example 3.1.1. Remember, only one of those descriptions actually defined a set, that being the set of “Major League Baseball players who hit at least 300 home runs in the 1990s”.

First, instead of repeatedly saying “the set of Major League Baseball players who hit at least 300 home runs in the 1990s”, we’ll use the capital letter M to refer to that set:

M = the set of Major League Baseball players who hit at least 300 home runs in the 1990s.

We can use lowercase letters to refer to specific players like so:

a	=	Joe DiMaggio
b	=	Ken Griffey Jr.
c	=	Babe Ruth
d	=	Mark McGwire
e	=	Mike Piazza
f	=	Barry Bonds
g	=	Rocky Marciano

We can go through and determine set membership for each of these players.

a ∉ M because Joe DiMaggio played baseball from 1936 to 1951, and as such didn’t play baseball in the 1990s, meaning he did not hit at least 300 home runs in the 1990s.

b ∈ M because Ken Griffey Jr. did hit at least 300 home runs in the 1990s. Even though he did play baseball for a brief time in 1989, as well as for a long time in the 2000s, that is irrelevant because we can still record how many home runs he hit in the 1990s all together, which was 382.

c ∉ M because Babe Ruth did not play baseball in the 1990s, and as such did not hit at least 300 home runs in the 1990s.

d ∈ M because Mark McGwire meets the requirement to be in M.

e ∉ M because even though Mike Piazza did play Major League Baseball in the 1990s, he did not hit enough home runs to be in M (he hit 240, 60 less than the required 300.)

f ∈ M because Barry Bonds hit at least 300 home runs in the MLB in the 1990s. As such, he meets the requirement for membership in M.

g ∉ M because Rocky Marciano did not hit at least 300 home runs in the MLB in the 1990s. As a matter of fact, Rocky did not play baseball professionally at all. Instead he was the Heavyweight World Champion Boxer, and was active from 1947 to 1951. So not only is this the wrong sport, but the time frame is also incorrect.

Let A denote the set

A = {1, 3, 5, 7, 9, 11, 13, 15, 17}.

Let B denote the set

B = {-12, -9, -6, -3}.

Since order and repetition don’t matter when building a set, we also have that

B	=	{-12, -9, -6, -3}
	=	{-3, -6, -9, -12}
	=	{-6, -12, -3, -9}
	=	{-12, -9, -6, -12, -3, -3}
	=	{-12, -3, -9, -12, -6, -12, -3, -12}

Since we explicitly state which elements are in each set, it is extremely easy to determine if an element is in either set simply by inspection.

3.14 ∉ A because it is not one of the listed numbers in A.

11 ∈ A because it is explicitly listed within the set’s definition.

-6 ∈ B because it is explicitly listed in B’s definition.

13 ∉ B because it is not one of the listed numbers in B.

Here, let A denote the set

A = {2, 4, 6, 8, 10, 12, …}

When defining set A, all the numbers we listed are positive even integers, hence we can reasonably assume that A refers to the set of all positive even integers. Of course, since there are infinitely many positive even integers, we have to use ellipses.

Let B denote the set

B = {1, 2, 4, 8, 16, 32, 64, …}

For B, it looks like all the numbers we listed are the non-negative integer powers of 2:

B = {2⁰, 2¹, 2², 2³, 2⁴, 2⁵, 2⁶, …}

Hence, we can reasonably assume that B refers to the set of all the powers of 2 where the exponent is a non-negative integer.

Consider the following set:

☒ = {1, 2, 4, 8, 16, …}

(Here, we use the symbol ☒ to refer to this set instead of a capital letter. The reason will be explained shortly.)

This almost looks like set B from Example 3.1.4, which was defined using powers of 2, hence we may think that ☒ is the set of all powers of 2 where the exponent is a non-negative integer.

Let’s reveal the next number in this particular set:

☒ = {1, 2, 4, 8, 16, 31, …}.

Hold on a second, 31 is not an integral power of 2! Hence, ☒ is not the set of all non-negative integer powers of 2.

So, what is it exactly? Is this set supposed be defined by some combination of arithmetic operators? If so, what are those operators? In what order are they applied?

As it turns out, this set describes the maximum number of regions a circle can be divided into by placing points around it’s circumference, and connecting those points with lines! Here are the next few numbers in the set:

☒ = {1, 2, 4, 8, 16, 31, 57, 99, 163, 256, …}.

There is a formula to calculate each number in the set:

However, this formula involves something called “binomial coefficients” or “combinations” which are what those numbers enclosed in parenthesis are. We’ll discuss those later in this book, but for now, it may appear to be complete gibberish.

The point is that, even though the first few terms of a listed set may seem to follow one pattern, there may be multiple patterns that fit, and so listing only a few numbers at the start may not adequately describe the desired set.

That’s why we used the ☒ symbol to denote this set: we wanted to make clear that this is a very misleading set of numbers at first, and as such is ambiguous. Since it’s ambiguous, we should not have described the set this way, hence using a box with an x in it to denote the set.

For anyone wanting to learn more about this set of numbers, this Wikipedia article is a great starting point.

There is another set of numbers that begins this way:

{1, 2, 4, 8, 16, 30, 57, 88, 163, 230, …}.

This set is also related to circles, but this set of numbers occurs when placing points equally around a circle’s circumference. The difference is that this set does not necessarily represent the maximum number of regions a circle can be divided into, such as the set of numbers above.

Never be ambiguous when describing a set!

Here is a typical example of the Set-Builder Method of defining a set:

A = { n | n is an even integer }.

Based off of this definition we see that

6 ∈ A	-12 ∈ A
100238 ∈ A	-12300450678 ∈ A

Of course, there are infinitely many other numbers in A. On the other hand,

1 ∉ A	-13 ∉ A	100343 ∉ A
3.1415 ∉ A	-2.718 ∉ A

among infinitely many other numbers.

Sometimes, multiple conditions can be placed on a set using the conjunction operator discussed in Chapter 1 like so:

B = { x | x is an integer ∧ x > 10 }.

So, any number that is simultaneously an integer, and larger than 10 is included in B. Thus we have that

90 ∈ B

92 ∈ B

100100 ∈ B

and so on. On the other hand, we have that

-2 ∉ B

10 ∉ B

0.4335 ∉ B

because -2 is not larger than 10, because 10 is not larger than itself, and because 0.4335 is not an integer nor is it larger than 10, so it fails both conditions.

It is possible to specify one condition before the vertical bar | in the set’s description like so:

C = { x is an integer | x ≤ -4 }.

Here, we are implying that, before we even consider what the normal condition is, we are presupposing that only integers can be included in C. In other words, this definition is equivalent to saying that

C = { x | x is an integer ∧ x ≤ -4 }.

Of course, we could use the disjunction operator as well, like so:

D = { x | x is a multiple of 0.4 ∨ x is a negative integer }

As such, we find that

2.4 ∈ D

-12 ∈ D

-100.4 ∈ D

because 2.4 is a multiple of 0.4 (or it’s a negative integer), because -12 is a negative integer and a (negative) multiple of 0.4, and because -100.4 is a multiple of 0.4 (or it’s a negative integer). However,

1 ∉ D

because 1 is neither a multiple of 0.4 or a negative integer.

Consider the following set:

X = { n is an integer | n² < 100 }

First, we know that we only have integers in X, because that is a condition for set inclusion.

The second condition is that the square of an integer must be less than 100. We can check a few numbers to be sure:

02	=	0	<	100
12	=	1	<	100
22	=	4	<	100
32	=	9	<	100
42	=	16	<	100
52	=	25	<	100

As a matter of fact we know that the first integer n where n² ≥ 100 is 10 since 10² = 100. As such, all the positive integers less than 10 are included in the set, so we have that

1 ∈ X	2 ∈ X	3 ∈ X
4 ∈ X	5 ∈ X	6 ∈ X
7 ∈ X	8 ∈ X	9 ∈ X

We also tested and verified that 0 ∈ X.

Since negative integers have positive squares, we also know that

-1 ∈ X	-2 ∈ X	-3 ∈ X
-4 ∈ X	-5 ∈ X	-6 ∈ X
-7 ∈ X	-8 ∈ X	-9 ∈ X

Any integer larger than 9, or less than -9 will not be included, because the smallest integer larger than 9 is 10, and 10² is not less than 100. Similarly, the largest integer smaller than -9 is -10, and (-10)² is also not less than 100.

As such, we know that all of the elements that are in X are the following:

X = {0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, -9}.

Even though we could list them, the Set-Builder Method at least gives us a rule to decide if some number is contained within the set. If we had simply listed the numbers, we wouldn’t know if there were some underlying condition, or if the numbers were simply chosen arbitrarily.

Of course, with a set like

Y = { n is an integer | n² ≤ 1000000000000 }

it would take a long time to fully list out the elements, unless we did something like this:

Y = {0, 1, -1, 2, -2, 3, -3, …, 999999, -999999, 1000000, -1000000}.

But we still prefer to work with the Set-Builder Method of describing Y because the Set-Builder Method is typically more succinct.

Consider the following set:

Ψ = { n³ | n is an integer ∧ n² < 100 }.

Here, we use the capital Greek letter psi Ψ just to add some variety to the types of names we’ve been using for our sets.

It looks like Ψ consists of perfect cubes, but one of the conditions is that the square of the underlying integer must be less than 100. Let’s examine a short list of some of the integers. In the following table, we’ll color all of the cells in the n² column that satisfy the requirement in green, and we’ll color red all those that do not.

n	n2	n3
-10	100	-1000
-9	81	-729
-8	64	-512
-7	49	-343
-6	36	-216
-5	25	-125
-4	16	-64
-3	9	-27
-2	4	-8
-1	1	-1
0	0	0
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	729
10	100	1000

Any integer less than -10 will have a square greater than 100, as will any integer greater than 10, so we don’t need to examine any more integers.

Remember that Ψ does not consist of those perfect squares, but instead consists of the corresponding cubes, which are listed in the right-most column of the above table, hence we know that

Ψ = { 0, 1, -1, 8, -8, 27, -27, 64, -64, 125, -125, 216, -216, 343, -343, 512, -512, 729, -729 }.

Consider the following sets:

A	=	{ x3 | x is an integer and |x3| < 100 }
	=	{ 1, -1, 8, -8, 27, -27, 64, -64 }
B	=	{ x2 | x is an integer and x2 < 100 }
	=	{ 1, 4, 9, 16, 25, 36, 49, 64, 81 }
C	=	{ 2n | n is an integer }
		{ 0, 2, -2, 4, -4, 6, -6, 8, -8, 10, -10, … }

We see that both A and B are finite sets, so we can see that they have the following cardinalities:

We see that C is an infinite set, so we do not speak of it’s cardinality.

Consider the set

X = { a, b, c, 1, 2, 3, x, y, z, {1, 2, 3}, {{1}, 2, 3}, {a} }.

This is a set with a wide variety of different types of objects. There are numbers, letters from the English alphabet (the a, b, c, x, y, and, z are not referring to variables in this particular case, just the letters themselves.), and even a few sets!

Let’s list out all of the elements contained within X in a table:

a	1	x	{1, 2, 3}
b	2	y	{{1}, 2, 3}
c	3	z	{a}

Notice that the element a is not the same thing as the element {a}: one is simply a letter, and the other is a set (containing the letter), and so are different elements entirely.

The same is true for the elements 1, 2, 3, {1, 2, 3}, and {{1}, 2, 3}. The element {1, 2, 3} is a set (though it contains many elements itself, the set only counts as one element for X.)

As such, there are no repeated elements in X, and since all are different, they all count towards the cardinality of X, meaning that we have

|X| = 12.

Of course the set {1, 2, 3} has its own cardinality:

| { 1, 2, 3 } | = 3.

But it is still 1 set, and it only counts for 1 element when considering the cardinality of X.

Similarly, we have that

| { {1}, 2, 3 } | = 3.

Contrast the above set with the following:

| { {1}, {2, 3} } | = 2

because the 2 and 3 are included in a single set within { {1}, {2, 3} }.