[Understanding Analysis by Stephen Abbott] - [Chapter 1 The Real Numbers] - [1.2 Some Preliminaries]

Intuitively speaking, a set is any collection of objects. These objects are referred as the elements of the set. For our purpose, the sets in question will most often be sets of numbers, although we will also encounter sets of functions and, on a few occasions, setes whose elements are other sets.

Give a set \(A\), we write \(A \in B\) if \(x\) (what ever if may be) is an element of \(A\). It \(x\) is not an element of \(A\), we write \(A \notin B\). Given two sets \(A\) and \(B\), the union is written \(A \cup B\) and is defined by asserting that $$ \text{\(x \in A \cup B\) provided that \(x \in A\) or \(x \in B\) (or potentially both)} $$ The intersection \(A \cap B\) is the set defined by the rule $$ \text{\(x \in A \cap B\) provided that \(x \in A\) and \(x \in B\)} $$

The set \(\emptyset\) is called the empty set and is understood to be the set that contains no elements. An equivalent statement would be to say that \(E\) and \(S\) are disjoint.

The inclusion relationship \(A \subseteq B \) or \(B \supseteq \ A\) is used to indicate that every element of \(A\) is also an element of \(B\). In this case, we say \(A\) is a subset of \(B\), or \(B\) contains \(A\). To asset that \(A = B\) means that \(A \subseteq B\) and \(B \subseteq A \). Put another way, \(A\) and \(B\) have exactly the same elements.

As mentioned, most of the sets we encounter will be sets of real numbers. Give \(A \subseteq R\), the complement of A, written \(A^\complement\), refers to the set of all elements of \(R\) not in \(A\). Thus, for \(A \subseteq R\), $$ \text{\(A^\complement\) = {\(x \in R\) : \(x \notin A\)}} $$ De Morgan's Laws states that $$ \text{ \((A \cap B)\complement = A^\complement \cup B^\complement\) and \((A \cup B)\complement = A^\complement \cap B^\complement\) } $$

Definition 1.2.3 Given two sets \(A\) and \(B\), a function from \(A\) to \(B\) is a rule or mapping that takes each element \(x \in A \) and associate with it a single element of \(B\). In this case, we write \(f : A \to B\). Given an element \(x \in A\), the expression \(f(x)\) is used to represent the element of \(B\) associated with \(x\) by \(f\). The set \(A\) is called the domain of \(f\). The range of f is not necessarily equal to \(B\) but refers to the subset of \(B\) by \(\text{\{\(y \in B\) : \(y = f(x)\) for some \(x \in A\)\} }\).

Triangle Inequality. The absolute value function is so important that it merits the special notation \(|x|\) in place of the usual \(f(x)\) or \(g(x)\). It is defined for every real number via the piecewise definition $$ |x| = \begin{cases} x, & \text{if $x \geqslant 0 $ } \\ -x, & \text{if $x < 0$ } \end{cases} $$ With respect to multiplication and division, the absolute value functions satisfies $$ |ab| = |a||b| \\ |a + b| \leqslant |a| + |b| $$ Verifying these properties is just a matter of examining the different cases that arise when \(a\), \(b\) and \(a + b\) are positive and negative. The second property is called triangle inequality.

Given three real numbers \(a\), \(b\), and \(c\), we have $$ |a - b| = |(a - c) + (c - b)| $$ By the triangle inequality $$ |(a - c) + (c - b)| \leqslant |a - c| + |c - b| $$ so we get $$ |a - b| \leqslant |a - c| + |c - b| $$ Now, the expression \(|a - b|\) is equal to \(|b - a|\) and is best understood as the distance between the points \(a\) and \(b\) on the number line. With this interpretation, we know why it is called triangle inequality.

Writing rigorous mathematical proofs is a skill best learned by doing, and there is plenty of on-the-job training just ahead. As Hardy indicates, there is an artistic quality to mathematics of this type, which may or may not come easily, but that is not to say that anything especially mysterious is happening. A proof is an essay of sorts. It is a set of carefully crafted directions, which, when followed, should leave the reader absolutely convinced of the truth of the proposition in question. To achieve this, the steps in a proof must follow logically from previous steps or be justified by some other agreed-upon set of facts. In addition to being valid, these steps must also fit coherently together to form a cogent argument.

The one proof we have seen at this point uses an indirect strategy called proof by contradiction. This powerful technique will be employed a number of times in our upcoming work. Nevertheless, most proofs are direct. (It also bears mentioning that using an indirect proof when a direct proof is available is generally considered bad form.) A direct proof begins from some valid statement, most often taken from the theorem's hypothesis, and then proceeds through rigorously logical deductions to a demonstration of the theorem's conclusion. An indirect proof always begins by negating what it is we would like to prove. The argument then proceeds until a logical contraction with some other accepted fact is uncovered. Many times, this accepted fact is part of the hypothesis of the theorem. When the contradiction is with the theorem's hypothesis, we technically have what is called a contrapositive proof.

Theorem 1.2.6 Two real numbers \(a\) and \(b\) are equal if and only if for every real number \(\epsilon > 0\) if follows that \( |a - b| < \epsilon\)

There are two key phrases in the statement of this proposition that warrant special attention. One is "for every", the other is "if and only if". To say "if and only if" in mathematics is an economical way of stating that the proposition is true in two directions. In the forward direction, we must prove the statement

if \(a = b\), the for every real number \(\epsilon > 0\) if follows that \(|a - b| < \epsilon\)

We must also prove the converse statement

If for every real number \(\epsilon > 0\) it follows that \(|a - b| < \epsilon\), then we must have \(a = b\)

For the proof of the first statement, there is really not much to say. if \(a = b\), then \(|a - b| = 0\), and so certainly \(|a - b| < \epsilon\) no matter that \(\epsilon > 0\) is chosen.

For the second statement, we give a proof by contraction. The conclusion of the proposition in this direction states that \(a = b\), so we assume that \(a \neq b\). Heading off in search of a contradiction brings us to a consideration of the phrase "for every \(\epsilon\)." Some equivalent ways to state the hypothesis would be to say that "for all possible choices of \(\epsilon > 0\)" or "no matter how \(\epsilon > 0\) is selected, it is always the case that \(|a - b| < \epsilon\)." But assuming \(a \neq b\), the choice of $$ \epsilon_0 = |a - b| > 0 $$ poses a serious problem. We are assuming that \(|a - b| < \epsilon\) is true for every \(\epsilon > 0\), so this must certainly be true for the particular \(\epsilon_0\) just defined. However, the statements $$ \text{\(|a - b| < \epsilon_0\) and \(|a - b| = \epsilon_0\)} $$ cannot both be true. This contradiction means that our initial assumption that \(a \neq b\) is unacceptable. Therefore, \(a = b\), and the indirect proof is complete.

One of the most fundamental skills required for reading and writing analysis proofs is the ability to confidently manipulate the quantifying phrases "for all" and "there exists."