Real Analysis: Math 4050-4060

A statement is any sentence that must be either true or false. So “5 is a natural number” is a true statement and “New York is the capital of the United States” is a false statement. But “Who are you?” is not a statement because it makes no sense to ask if it is true or false, and “onions taste good” is not a statement because whether it is true or false depends on who we ask. We will often use symbols with special font, such as \(\cal P \) or \(\cal Q \text{,}\) followed by a colon, to denote statements. So we could write “\(\cal P\) : 5 is a natural number” or “\(\cal Q\) : New York is the capital of the United States”. If we write \({\cal P}(x)\text{,}\) it means that the statement \(\cal P\) depends on whatever \(x\) may be. So for example “\({\cal P}(x)\text{:}\) \(x\) is a natural number less than 5” is a statement that is true if \(x\) is 1,2,3 or 4, but it is false otherwise. In this situation we call \(x\) a variable. Of course a variable need not represent a number. We could consider the statement “\({\cal P}(x)\text{:}\) \(x\) is a student in this class”, and then \(x\) represents a person.

We will now review some logic. If \(\cal P\) is a statement, we denote by \(\sim \cal P\) its negation. So \(\sim \cal P\) is true when \(\cal P\) is false, and \(\sim \cal P\) is false when \(\cal P\) is true. For example, if \({\cal P} (x)\text{:}\) “\(x\) is an integer greater than 1” then \(\sim {\cal P} (x)\text{:}\) “\(x\) is an integer less than or equal to 1”.

If \(\cal P\) and \(\cal Q\) are statements, we can construct another statement, called an implication denoted by \({\cal P} \Rightarrow {\cal Q}\) that is false only when \(\cal P\) is true and \(\cal Q\) is false. In all other cases it is true. We read this as "\(\cal P\) implies \(\cal Q\)", or "If \(\cal P\text{,}\) then \(\cal Q\text{.}\)" The logic is that we are thinking that if \(\cal P\) is true, then \(\cal Q\) must also be true, or that \(\cal Q\) follows from \(\cal P\text{.}\)

To understand why we say that the statement \({\cal P} \Rightarrow {\cal Q}\) is always true except in the case that \(\cal P\) is true and \(\cal Q\) is false, think of it as a promise. So suppose \(\cal P\) is the statement “the New Orleans Saints win the game” and \(\cal Q\) is the statement “Dustin will take Beverly out to dinner”. Then \({\cal P} \Rightarrow {\cal Q}\) is the promise that if the Saints win the game, Dustin will take Beverly out to dinner. How can this promise be broken? Only if the Saints win, and yet Dustin does not take Beverly out to dinner. In all other cases, the promise will not be broken. So if the Saints lose, the statement \({\cal P} \Rightarrow {\cal Q}\) remains true for sure, whether or not Dustin takes Beverly out to dinner, because there is no broken promise.

Note that the statement “There are some people who don’t like chocolate” is the precise negation of the statement “Everybody likes chocolate”. This is reflected by the fact that \(\forall \) turns to \(\exists \) and \({\cal P}\) turns to \(\sim {\cal P}\text{,}\) and provides a useful rule of thumb to write the negation of complex logical statements: change all \(\forall \) to \(\exists \) and all \(\exists \) to \(\forall \text{,}\) and change every statement to its negation.

We will review some of the basic notions of set theory. A more complete treatment of the theory can be found in many freely avaialble online sources such as this textbook on Discrete Mathematics¹.

As we already mentioned, we need to be precise when making definitions, and we should only use concepts that have already been previously defined. But we need to start somewhere. There are notions that must be treated as understood, without appealing to any other previously defined concepts. Basic examples of these notions are those of set² and membership. A set is a collection of elements. An element is a member of a set (or it belongs to it, or is contained in it) if it is one of the elements that make up the set. These notions are primitive, in the sense that they cannot be explained or defined in terms of other notions. We need to start from here. We will use curly braces \(\{ \ldots \}\) to represent what a set contains, as we have already done in the notation for \(\mathbb N\text{,}\) \(\mathbb Z\) and \(\mathbb Q\) in the previous section.

So \(\mathbb N\text{,}\) \(\mathbb Z\) and \(\mathbb Q\) are basic examples of sets. If \(A\) is a set, we will use the notation \(x\in A\) to mean that \(x\) is an element of \(A\text{,}\) or that \(x\) belongs to \(A\text{,}\) and \(x\not \in A\) to mean that \(x\) is not an element of \(A\text{.}\) So for example \(3\in {\mathbb N}\) and \(\frac{1}{2}\not \in {\mathbb Z}\text{.}\)

Once we feel comfortable with the notion of a set and what it means for something to belong to it, we can define the basic set operations such as union and intersection. This gives us the opportunity to illustrate our first definition.

Note that the word “or” in math is used in a different way than it is used in ordinary language. In math, \(\cal P\) or \(\cal Q\) means either \(\cal P\text{,}\) or \(\cal Q\text{,}\) or both. So if \(x\) is in \(A\cup B\text{,}\) then \(x\) could be in \(A\text{,}\) or in \(B\text{,}\) or in both \(A\) and \(B\text{.}\) Compare with the ordinay use of "or" in the sentence “Would you like tea or coffee?”. In this sentence it is evident that the choice is understood to be between one or the other, not both.

Let’s pause a moment to examine what the above definition of union and intersection does. First of all, it uses plain English. Then, it does two different things: it explains what the new concepts are, and it assigns a name to them. The assigned names are union and intersection. Then, it also introduces some notation in order to efficiently use the new concepts in writing. The notation introduced consists of the two symbols \(\cup\) and \(\cap\) that represent union and intersection, respectively. In summary, a definition often does three things: it explains the meaning of a new concept; it reserves a word or phrase to be used for it; and (sometimes) it also reserves a symbol to be used for it.

We stress the fact that plain English was used. This could have been done in a different way. We could have written the definition as follows:

This second way to define union and intersection is perfectly good and correct. But, in this course we will prefer to use plain English language whenever possible, in definitions, in statements of theorems, and in proofs. This does not mean that we will not use symbols. On the contrary, symbols are very often essential to carry out complex proofs. But they should not be a substitute for words when the concept is more clearly explained or understood without them. Most people who have never heard of union and intersection will more easily understand their definition in the one with the words rather than in the one with the symbols. It is often useful, as an exercise, to express in words concepts presented using only symbols, and likewise to write in compact symbolic form statements that were made using only words.

In our discussion of sets so far we have not addressed how large a set can be. Of course we already know that there are infinitely many numbers, so the sets \(\mathbb N, Z, Q\) are all infinite. But sets such as \(\{n\in {\mathbb N}: n\leq k\}\text{,}\) where \(k\) is a positive integer, are clearly not infinite, in fact this set has exactly \(k\) elements. A set that is not infinite is said to be finite. For such a set, we define its cardinality to be the number of its elements, and denote the cardinality of the set \(A\) by \(|A|\text{.}\) So for example if \(A=\{1,3,5,7,9\}\) then \(|A|=5\text{,}\) and \(|\{n\in {\mathbb N}: n\leq k\}|=k\text{.}\)

For example, \(\mathbb Z \setminus \mathbb N\) is the set of all negative integers, together with 0, and \(\mathbb Q \setminus \mathbb Z\) is the set of rational numbers that, when written in lowest terms, have a denominator larger than 1.

Sometimes we consider all our sets as being subsets of some fixed set, that we call the universal set. So for example, if we work with positive integers only, the universal set is \(\mathbb N\text{.}\)

We can now state some basic relationships between the complement and the set operations of union and intersection.

In particular, the Cartesian product of a set \(A\) with itself is the set \(A\times A\) of all pairs \((a,b)\) where \(a,b \in A\text{.}\) Two important notions are associated with \(A\times A\text{.}\)

Assuming we already know what \(a\lt b\) means, an important example of a binary relation is \(R=\{(a,b): a\lt b\}\text{.}\) In fact, in the next chapter, we will use the notion of binary relation to define what \(a\lt b\) means.

The usual operations of addition and multiplication are basic examples of binary operations. Instead of using the notation \(f(x,y)\) we use \(x+y\text{,}\) or \(xy\text{,}\) but that is the only difference: both addition and multiplication are just functions with domain \(A\times A\) and co-domain \(A\text{,}\) where \(A\) can be \(\mathbb N\text{,}\) or \(\mathbb Z\text{,}\) or \(\mathbb Q\text{.}\)