Foundations of Mathematics: Basics and Vocabulary Explained

Basics and Vocabulary The goal of this hour is to give you the basic vocabulary you need to understandmore advanced material later in your mathematics studies. We begin with thedefinition of a set, then discuss a fancy word called cardinality, which is really just afancy way of saying "size." We then go over two ways to combine sets—intersectionand union. All this will be presented in a dry manner at first, but there is light at theend of the tunnel. So, what is a set? The term set refers to a collection of objects, called its elements. Sets are denotedusing braces or brackets and it is possible to have elements that are themselvessets. For example, the set A can be written as {a}, whereas the set E could bewritten as {apple, monkey, DE}. Sets are collections of elements. In this example, A is a set with four elements: theelement one, the element two, the element minus three and the element seven. E isa set with three elements: apple, monkey and DE. The elements in a set can beessentially anything. We use braces to enclose sets, so that we can see which thingsare contained within them. For example, when we write two inside A – without thebraces – that means two is an element of A. Minus three is also an element of A;likewise for eight, which is not an element of A. So we write "not in" using this symbolhere — it means exactly what it says on the tin: eight is not in A. The last notion we'll talk about is that of cardinality. Cardinality is the size or numberof elements in a set. When we refer to the absolute value of a set, we use thisnotation: |A|. So in this particular case, where A is equal to one, two, minus three,seven, the cardinality of A is four, and the cardinality of E is three. That's all there isto it! Let us move on to the next topic. We are writing three sets for you here. A is equal toone, two, minus three, seven – as you have seen before. Let us say B is equal totwo, eight, minus three, ten, and D is equal to five and ten. Now you will notice thatof these three sets, they share some elements in common (the intersection), someelements they do not share in common (the union). There are two concepts calledintersection and union which really allow us to talk about those more rigorously. First, let's consider the intersection of sets A and B. The intersection of these sets isthe set of elements that are contained in both A and B. In other words, it is the set ofelements shared by both sets. To determine the intersection of sets A and B, wemust first determine what elements these sets have in common. We see that two iscontained in both A and B, as well as minus three. Since these are the only elementscontained in both sets, their intersection equals {2, -3}. Next let us work out the

intersection between set B and set D. This intersection is defined to be the set ofelements that both B and D contain in common; however there are none so this setis empty or { }. The notation for an empty set is a boldface zero with a slash throughit: 0/. This is called the empty set. Hard to spell. And by convention, we always say that the cardinality of the empty set is zero – there's nothing in it. Here is a simple example of a set that can be defined in multiple ways. Another wayof writing A intersect B is to give you a recipe for computing it yourself. So the

definition of A intersects B, instead of listing out the elements, we can list it this way:it's a set of x, we don't know what x is, but now I'm gonna give you conditions that xsatisfies, that's what this little colon here means. The set of x such that x is in A andx is in B. This notation here is very, very important and we're gonna see it over andover again. You can think of "A intersect B '' like membership in a club. The club has two rules:(1) you must be in A and (2) you must be in B. If any x satisfies both conditions, thenthe bouncer will let x into the club. If no x satisfies both conditions, then no x getsinto the club. For example, suppose two people come along and say: "I want to getinto this music club." The bouncer checks his ID and says "okay"; so two get in.Suppose one comes along: "I want to get into this music club." The bouncer checkshis ID and says "no"; so one does not get into the club. Let's rewrite those sets and give you a different definition. So A is equal to one, two,minus three, seven; B is equal to 2, 8, -3, 10; and D is equal to 5, 10. The next ideawe're gonna define here is the idea of the union. A union B is read as the set ofelements that are in A or in B or in both. The symbol for union is a rectangle with asmaller rectangle inside it. When you see A union B, think about it as being inclusiveof elements from A and from B. In this example, you have 1, 2 and 5 from set A and7 from set B. So the answer is 8, 10. In another notation, A union D would mean 1, 2,3, 7 and 10.