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Basics of Set Theory

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Basics of Set Theory 2.2. THE BASICS OF SET THEORY 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.
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