In this section we explain the importance of set theory and introduce its basic concepts and definitions. We also show the notation with sets that has become standard throughout mathematics.
The Importance of Set Theory
One striking feature of humans is their inherent propensity, and ability, to group objects according to specific criteria. Our prehistoric ancestors were hunter-gatherers who grouped tools based on their survival needs. They eventually settled and formed strict hierarchical societies where a person belonged to one class and not another. Today, many of us like to sort our clothes at home or group the songs on our computer into playlists.
As we look at the accomplishments of science at the dawn of this new millennium, we can point to many impressive classifications. In chemistry there is Mendeleev’s periodic table, which lists all the known chemical elements in our universe and groups them based on common structural characteristics (alkaline metals, noble gases, etc.) In biology there is now a vast taxonomy that systematically sorts all living organisms into specific hierarchies: kingdoms, orders, phyla, genera, etc. In physics, all the subatomic particles and the four fundamental forces of nature have now been classified under an incredibly complex and refined theory called the Standard Model. This model – the most accurate scientific theory ever devised – surely ranks as one of mankind’s greatest achievements.
The idea of sorting out certain objects into similar groupings, or sets, is the most fundamental concept in modern mathematics. The theory of sets has, in fact, been the unifying framework for all mathematics since the German mathematician Georg Cantor formulated it in the 1870’s. No field of mathematics could be described today without reference to some kind of set, or be practiced without the use of set notation. A geometer, for example, may study the set of parabolic curves on a plane or the set of spheres in a variety of different spaces. An algebraist may work with a set of equations or a set of matrices. A statistician typically works with large data sets. Even numbers, as it turns out, are formally described in terms of sets. The most important unresolved problem in mathematics at the moment deals with the set of prime numbers. This problem in number theory is known as Riemann’s Hypothesis, named after the great German mathematician Bernhard Riemann (1826 – 1866), and a large chunk of mathematics existing now depends on it being true (while there is no reason to think the conjecture is actually false, no one has been able to prove it since Riemann formulated it in 1859). The Clay Institute in Cambridge, MA, will award a million dollars to whoever solves it1.
More broadly, the concept of set membership, which lies at the heart of set theory, explains how statements with nouns and predicates are formulated in our language – or any abstract language like mathematics. Consider the following statement:
Batman is a caped superhero.
This sentence is essentially stating that Batman belongs to the set of superheroes who wear capes (alongside Superman, say, but not Spiderman). This important link between predicates and sets will be discussed again in the next chapter. For now, let us conclude by reiterating that set theory is intrinsically connected to logic and, as a result, serves as the foundation for all mathematics.
What is a Set?
Everyone has an intuitive understanding of sets. They are ubiquitous to our daily life. We use sets of bed linen in our bedrooms, carry around a set of keys, tee off with a set of golf clubs, and play poker with a set of cards. In elementary school we learn basic arithmetic with the set of counting numbers and learn to read words assembled from the set of letters in our alphabet.
DEFINITION: A set is a collection of distinct objects, none of which is the set itself. The objects in a set are called the elements, or members, of the set.
The elements of a set could be anything conceivable, provided they are unambiguous entities. For example, a set can consist of numbers, shapes, letters, colors, golf clubs, marine fossils, U.S. senators, tennis players, chromosomes, or even sets themselves.
The way in which the elements of a set are listed is unimportant. In other words, there is no prescribed order for the placement of elements in a set. For this reason, sets can be visualized as receptacles (such as a bin or a basket) into which elements are to be thrown indiscriminately. For example, the set of numbers that are equal to their square consists of 0 and 1 (you can check this algebraically by solving the equation 푥 = 푥2, which implies 푥2 − 푥 = 0 ⇒ 푥(푥 − 1) = 0). This set can also be described as containing 1 and 0, listing the two numbers in reverse order. Either way,
1 The Riemann Hypothesis is one of the seven millennium problems announced in 2000 by the Institute. As of now, the only problem in the list that has been solved is Poincaré’s Conjecture, named after the great French mathematician Henri Poincaré (1854 – 1912). This important result in topology, an advanced field of geometry, was proved in 2003 by the Russian mathematician Grigori Perelman (born 1966). the solution set remains unchanged. Another example is when someone places an order at the drive-thru window of a fast food restaurant. Whether that person yells at the intercom “I’d like a cheeseburger, a soda, and fries” or “I’d like a soda, fries, and a cheeseburger,” or any other combination of the three items, does not change the fact that this person expects to receive a meal consisting of a cheeseburger, fries, and a soda!
Since the elements of a set are distinct, some elements may be listed repeatedly 1 2 2 4 without consequence. So, for instance, the set containing the fractions , , , is in fact 1 1 2 2 the set containing the numbers 1 and 2.
Lastly, none of the objects of the set can be the set itself. We discard this possibility to avoid running into Russell’s Paradox, a famous problem in mathematical logic discovered by the great British logician Bertrand Russell (1872 – 1970) in 1901. We’ll discuss this paradox later in this section.
Set Notation
We write sets using braces and denote them with capital letters. The most natural way to describe sets is by listing all its members. This is called the roster method.
EXAMPLES: . 퐴 = {1,2,3, … ,10} is the set of the first 10 counting numbers, or natural numbers. . 퐵 = {푅푒푑, 퐵푙푢푒, 푌푒푙푙표푤} is the set of primary colors. . ℕ = {1,2,3,4, … } is the set of all natural numbers. . ℤ = {… , −3, −2, −1,0,1,2,3, … } is the set of all integers. Note the use of the ellipsis “…” to describe the infinite listings in the number sets ℕ and ℤ.
We use the Greek letter epsilon (∈) to denote set membership. Thus we write “푥 ∈ 퐴” to mean “the object 푥 belongs to the set 퐴” and “푥 ∉ 퐴” to mean “the object 푥 does not belong to the set 퐴”.
EXAMPLES: Using the sets defined previously, we could write all the following: 1 ∈ 퐴, 12 ∉ 퐴, 2 12 ∈ ℕ, 0 ∉ ℕ, 0 ∈ ℤ, ∉ ℤ, or Black ∉ 퐵. 3
Since many sets cannot be described by the roster method (this can prove to be impossible, even with the use of ellipses), we resort to the more abstract and more powerful set-builder, or predicate, notation. In that notation we write the set according to what types of objects belong to the set (these are placed to the left of the “|” symbol, which means “such that,” inside the braces) as well as the conditions that these objects must satisfy in order to belong to the set (these are placed to the right of the “|” symbol inside the braces).
EXAMPLES: . The set of rational numbers, or fractions, which is denoted by ℚ, cannot be described using the roster method. Instead, we write ℚ in set-builder notation 푝 as follows: ℚ = { |푝, 푞 ∈ ℤ 푎푛푑 푞 ≠ 0}, or “ℚ is the set of all fractions with 푞 numerator 푝 and denominator 푞, such that 푝 and 푞 are integers and 푞 is not zero.” . We could write set 퐴 in our previous example as {푥|푥 is a natural not greater than 10}, or 퐴 = {푥|푥 ∈ ℕ 푎푛푑 푥 < 11}. . The set 퐷 = {6,12,18,24,30}, which consists of the first five natural multiples of 6, could also be written as 퐷 = {6푛|푛 = 1,2,3,4,5}.
Well Defined Sets
There is a famous problem in analytic philosophy called the sorites (or heap) paradox. The problem is an illustration of a class of paradoxes that arise from the use of vague predicates. Consider a heap of sand. What exactly defines such a heap? Suppose the heap contains around ten million grains of sand. Does removing a single grain of sand preserve the heap? Surely yes. What if another grain of sand is removed? Still, the heap would remain in place virtually untouched. In fact, the same conclusion would hold for thousands and thousands of additional grain removals. Now, imagine you continue this procedure, one grain at a time, until only a few grains of sand are left in the heap. Would you still call a collection of, say, 10 or 11 grains of sand a heap? Surely not. The question is: when did the heap become a non-heap? There is no real way to answer this question. This is a problem in language that stems from the vagueness of the predicate is a heap. In terms of sets, we avoid this type of problem by making sure the criteria for membership in a set are well defined. This leads us to the next definition.
DEFINITION: A set is said to be well defined if it is unambiguous which elements belong to the set.
In other words, if 퐴 is a well defined set, then the yes or no question “Does 푥 belong to 퐴?” can always be answered for any object 푥.
EXAMPLES: . All the sets introduced previously (퐴, 퐵, ℕ, ℤ, ℚ) are well defined since it is clear which objects belong to each one of them. . The set of large numbers is not well defined since it is unclear which numbers should be considered large. Does the number 210 belong to this set? How about 2010 or 201710? . The set of numbers with 20 or more digits (where the leading digit is not zero) is well defined. Note that if we choose to call these particular numbers large, then the previous set becomes well defined. . The set of words in the English language is not well defined since there are different standards of dictionary definitions used for words. Moreover, it is not clear whether homographs (words that are spelled the same but have different meanings) should be counted only once or multiple times (e.g. “bass” could be a low, deep voice or a type of fish). . The set of great New York Mets players or the set of nice students in a logic class are not well defined since membership to these two sets is bound to be subjective. Indeed, what makes a player great or a student nice?
Besides the problem of vague criteria for set membership, the concept of a well defined set also became necessary after Russell asked in 1901 the following question: “Suppose U is the set of all sets that do not contain themselves. Does U belong to itself?” Think about this question carefully. If you answer yes, then that’s a contradiction since the set U cannot, by definition, contain itself. If you answer no, this is also a contradiction since U should then contain itself by definition! Russell used a colloquial version of this rather technical question to illustrate the problem. He called it the barber paradox. Suppose a town barber is the person who shaves all people in the town who do not shave themselves – and only those people. Then who shaves the barber? Again, either yes or no answer leads to a logical contradiction. This nagging paradox rocked the world of mathematical logic at the time. However, it was eventually resolved by Russell himself using his so-called theory of types and then later by Zermelo using an axiomatic method. Today, Zermelo’s solution is preferred to Russell’s. At the simplest level, working with well-defined sets safely bypasses this problem.
Number Sets
Here are some important number sets used in mathematics:
. ℕ = {1,2,3,4, … } is the set of counting numbers, or natural numbers. . 푊 = {0,1,2,3,4, … } is the set of whole numbers. . 푃 = {2,3,5,7,11,13, … } is the set of prime numbers2. . ℤ = {… , −3, −2, −1,0,1,2,3, … } is the set of integers. 푝 . ℚ = { |푝, 푞 ∈ ℤ 푎푛푑 푞 ≠ 0} 푞 is the set of rational numbers, or fractions. This set can also be described as containing all terminating (e.g. 1.23) or non-terminating but repeating decimals (e.g. 1.232323…). . ℚ̅ is the set of irrational numbers. This set can also be described as containing all non-terminating and non-repeating decimals (e.g. 1.2323323332…). Some of the most important numbers in mathematics belong to this set, including 휋, √2, 푒 (the natural base), and 휙 (the golden ratio). . ℝ is the set of real numbers. These are all the numbers that can be placed on a one-dimensional number line extending with no end on both the negative and positive sides, such as the x- or y-axis in the Cartesian plane. Combining the rational numbers with the irrational numbers produces all the real numbers.
Set Equality
DEFINITION: Two sets 퐴 and 퐵 are said to be equal (denoted, as usual, by 퐴 = 퐵) if and only if both sets have the exact same elements.
EXAMPLES: . The set of even naturals is given by 퐸 = {2,4,6,8, … } = {2푥|푥 ∈ ℕ}. . The set of states in the continental U.S. that border the Pacific Ocean and the set {Washington, Oregon, California} are equal. . The unit interval (0,1), which consists of all the real numbers between 0 and 1, is also equal to the set {푥 ∈ ℝ|0 < 푥 < 1}. . The set of squares is equal to the set of equilateral rectangles. . The set of even prime numbers is equal to {2} since 2 is the only prime number that is even.
The if and only if logical connective in the definition means that both parts of the statement (“퐴 = 퐵” and “both sets have the exact same elements”) are interchangeable. Logically speaking, this means that each part of the statement implies the other. This connective, called a biconditional, will be discussed in the next chapter.
2 Prime numbers are all the naturals greater than 1 that are only divisible by themselves or 1. These numbers play a fundamental role in number theory. They are further discussed in Chapter 5. Since it doesn’t matter how the elements of a set are listed, or how many times an element is repeated in a set, set equality can lead to peculiar writings. For example, {2,4,6,8} = {4,8,6,2} = {2,4,2,8,6,4,6,2}. It is, however, rare to see repeated elements in a set.
The Empty Set
A very important set is the empty set, or the null set, which is the set with no elements. The set of living mammoths, for example, is the null set. So is the set of numbers that are both odd and even, or the set of standard decks of cards with five aces. We denote the empty set by ∅, or { }. We could also write, for example, ∅ = {푥|푥 ∈ ℕ 푎푛푑 푥 < 0} or ∅ = {푥|푥 ∈ 퐴 푎푛푑 푥 ∉ 퐴}, where 퐴 is any set.
Note that we say the empty set and not an empty set. The reason why there is only one empty set is simple: if two sets 퐴 and 퐵 are empty, then they both have no elements. But that means they have the same exact elements, namely none! Therefore, 퐴 and 퐵 must be equal by our previous definition.
Cardinality of a Set
DEFINITION: The cardinality, or cardinal number, of a set 퐴 is the number of elements that belong to 퐴. We denote this number by 푛(퐴) or |퐴|.
EXAMPLES: . The cardinality of the set of digits 퐷 = {0,1,2, … ,8,9} is 10 so we write |퐷| = 10. . The cardinality of the set {2,4,6, … ,18,20} = {2푛|푛 = 1,2, … ,10} is also 10. . The cardinality of the set of letters in the English alphabet is 26. . The cardinality of the empty set is 0 since it contains no elements. So |∅| = 0. However, the cardinality of the set {0} is 1 since this set contains exactly one element: the number 0. So |{0}| = 1.
The cardinality of a set is then, in some sense, a measure of its “size”. For instance, the set of fingers and the set of vowels have the same “size” since the cardinality of both sets is 5. If the cardinality of a set is a whole number, then the set is said to be finite. Otherwise, the set is said to be infinite. Cantor came up with a precise way to define the concept of “size” for infinite sets. This allowed him to study the cardinality of infinite sets and led him to what he called transfinite numbers. We’ll discuss this more advanced topic later in the chapter.
Subsets and Proper Subsets
DEFINITION: If all the elements of a set 퐴 are also elements of another set 퐵, then 퐴 is called a subset of 퐵. We denote this by 퐴 ⊆ 퐵.
EXAMPLES: . All the male students at a college constitute a subset of the entire college student population. . The set {0,2,4,6,8} is a subset of 퐷, the set of digits. . The set of vowels is a subset of the set of letters in the English alphabet. . The set of women who are registered as independent voters is a subset of all registered U.S. voters. . Since even naturals are special types of natural numbers (those divisible by 2), subset of 퐸 is a subset of ℕ, or 퐸 ⊆ ℕ. . You can check that the number sets introduced earlier are all related as follows: 푃 ⊆ ℕ ⊆ 푊 ⊆ ℤ ⊆ ℚ ⊆ ℝ. For example, 푃 ⊆ ℕ since every prime number is a natural number and ℤ ⊆ ℚ since every integer is a fraction with denominator 1.
So to say that 퐴 is a subset of 퐵 is to say that 퐴 is contained within 퐵. If you think of sets A and 퐵 as lists, such as team rosters or name directories, then 퐴 is some part of the list 퐵, such as the starting players in a team or the people on the directory whose name starts with “P”. Note that this part could range anywhere between an empty list (i.e. 퐴 is the empty set) and the entire list 퐵 (i.e. 퐴 = 퐵). These two extreme cases are permissible under the definition of 퐴 ⊆ 퐵.
A common confusion is to mistake a subset with a member of a set. For example, the set {푎, 푒} is a subset of the set of vowels, but not a member of it. We then write {푎, 푒} ⊆ {푎, 푒, 푖, 표, 푢 } but not {푎, 푒} ∈ {푎, 푒, 푖, 표, 푢 }. This is not to say that individual vowels are members of this set. It is indeed true that 푎 ∈ {푎, 푒, 푖, 표, 푢 } and 푒 ∈ {푎, 푒, 푖, 표, 푢 }.
The idea of a subset is so ubiquitous to our human experience that many subsets are defined in our language by words that start with the prefix sub-, such as submarine, subatomic, subculture, subcontract, subtext, subway, and so on. In some cases, the concept of subset in the word is literal (e.g. subway) and in others it is figurative (e.g. subtext).
DEFINITION: If a set 퐴 is a subset of 퐵 and the two sets are not equal, then we call 퐴 a proper subset of 퐵. This is denoted by 퐴 ⊂ 퐵.
EXAMPLES: . The male students at Suffolk County Community College constitute a proper subset of the entire college student population. . The male students at an all-male college do not constitute a proper subset of the entire college student population. . The set 0,2,4,6,8 is a proper subset of D, the set of digits. . The set of vowels is a proper subset of the set of letters in the English alphabet. . The set of women who are registered as independent voters is a proper subset of all registered U.S. voters. . The set 퐴 = {푎, 푏, 푐, 푑, 푒} is a subset of the set 퐵 = { 푙|푙 is one of the first 5 letters of the English alphabet}, but 퐴 is not a proper subset of 퐵 since 퐴 = 퐵. We then write 퐴 ⊆ 퐵 but not 퐴 ⊂ 퐵.
So to say that 퐴 is a proper subset of 퐵 is to say that 퐴 is contained within 퐵 and there must be at least one element in 퐵 that is not in 퐴.
Below are some important properties relating to subsets and proper subsets.
PROPERTIES: . Any set 퐴 is a subset of itself. Hence 퐴 ⊆ 퐴. This is clearly true . Less obvious is the fact that the empty set is a subset of any set 퐴 (i.e. ∅ ⊆ 퐴). You may think of this as stating that nothing is a part of anything. A better answer relies on the following proof by contradiction: start by assuming that this claim is false (so assume there exists a set 퐴 for which the empty set is not a subset). Then, by definition, the empty set must contain an element that is not in set 퐴. But this is absurd since the empty set is empty. Hence, the initial assumption was wrong and so ∅ ⊆ 퐴 for any set 퐴. . The empty set is a proper subset of any set 퐴, provided that 퐴 is not the empty set itself. . For finite sets 퐴 and 퐵, if 퐴 ⊆ 퐵 then |퐴| ≤ |퐵|. . For finite sets 퐴 and 퐵, if 퐴 ⊂ 퐵 then|퐴| < |퐵|.
Power Sets
DEFINITION: The power set of a set 퐴, denoted by 푃퐴, is the set consisting of all the distinct subsets of 퐴.
EXAMPLES: . The power set of the empty set if the set that only contains the empty set. So 푃∅ = {∅}. . If 퐴 = {훼, 훽}, then 푃퐴 = {{ }, {훼}, {훽}, {훼, 훽}}. . If 퐵 = {퐼, 퐼퐼, 퐼퐼퐼}, then 푃퐵 = {∅, {퐼}, {퐼퐼}, {퐼퐼퐼}, {퐼, 퐼퐼}, {퐼, 퐼퐼퐼}, {퐼퐼, 퐼퐼퐼}, 퐵}.
Based on the properties enumerated earlier, it is clear that the power set of a non-empty set 퐴 always contains at least two sets: the empty set and the set 퐴. A similar idea exists with natural numbers: a natural always has at least two factors: 1 and itself.
To see how power sets can be usefully applied to the theory of numbers, consider the question of finding all the factors of a number like 910. Some factors like 2, 5 and 10 might be easy to determine since the number ends in a zero (which implies the number is divisible by 2, 5 and 10), but other factors like 65 and 182 might not be as obvious. One systematic procedure to answer this question starts with the prime factorization of 910: 910 = 2 × 5 × 7 × 13. By looking at all the subsets of the set of factors {2,5,7,13} of 910, we can see all different products that factor into this number. For instance, the subset {5,13} corresponds to the factor 65 = 5 × 13 while the subset {2,7.13} corresponds to the factor 182 = 2 × 7 × 13. Here is a complete list of the 16 factors of 910: 1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910.
The following is an important result in set theory relating the cardinalities of finite sets to their power sets.
THEOREM: If 퐴 is a finite set with 푘 elements, then the power set of 퐴 has exactly 2푘 elements. Therefore we have:
| | |퐴| 푘 푃퐴 = 2 = 2 .
We include the proof of this theorem, which is done by mathematical induction, 2 3 in Appendix A. In our previous example, check that |푃퐴| = 2 = 4 and |푃퐵| = 2 = 8.
This result is applied in a variety of contexts, including many counting problems and procedures from number theory (check the question answered previously listing the 24 = 16 factors of 910). Here’s a classic example. Suppose your local pizzeria offers the following six toppings to put on their pie: onions, peppers, sausage, pepperoni, olives, and anchovies. A customer can order a pie that includes any combination of these toppings, ranging from the simplest one with no toppings (the “margherita”), to a pie with, say, only onions and peppers, to the one with all six toppings (the “supreme”). How many different pizza pies could you potentially order at this pizzeria? A systematic method to deduce this number would start with the “margherita,” then enumerate all the different pizzas that can be ordered with one topping, then all the different pizzas that can be ordered with two toppings, and so on until the “supreme” is reached. The answer turns out to be 64, which is also the sixth power of 2 ( 26 = 64). In other words, the question amounts to finding the cardinality of the power set of the set of toppings. We end this section by noting that one could also deduce this answer using other means. Another argument for this kind of problem will be shown in Chapter 7, where we turn to combinatorics – the mathematics of counting.