SIMPLE INTRODUCTION TO MUNBER THEORY

4. Munbers as Sets

4.1 Introduction. Sets into munbers

If numbers are easily the most useful mathematical construct ever conceived, sets must be the closest competitor. They may be built out of anything at all, even out of themselves. In Chapters 1 and 2, we built them out of munbers. Now, we want to take the reverse approach and build munbers out of sets, performing operations on them the way we would sets. In this chapter, we will take a look at various properties of sets and apply them to munbers.

DEFINITION. Let S be a set of numbers on F. Then the munber of S [M(S)] is defined as the munber a such that ∀ s ∈ F, s = a iff s ∈ S.

In other words, in order to change a set of numbers into a munber, one need only replace the braces on the set with double angle quotes.

NOTE. I will again refer to "numbers" and "munbers" in general terms in this chapter. See the Note at the beginning of Chapter 3 for my remarks on the use of fields.

4.2 Review of basic set operations

Recall that in Chapter 1, we defined intersections and unions of munbers. The intersection of two munbers is that munber containing only those numbers that are members of both munbers. If the munbers are disjoint, the intersection is undefined. The union is that munber containing only those numbers that are members of either or both munbers. Since a munber must have at least one member, a union is always defined.

We also defined the degree of a munber. If a is a munber, then the number of members of a is its degree, symbolized deg(a). This is an example of a function that maps a munber to a number [specifically, a positive integer]; there will be many more in this chapter.

THEOREM 4.1. INTERSECTION DEGREE THEOREM. Let a and b be munbers of finite degree. Then deg(a ∪ b) = deg(a) + deg(b) - deg(a ∩ b).

Proof. Let m = deg(a) and n = deg(b). In addition, let k = deg(a ∩ b). Then m + n is equal to the total number of members. But k of those members are the same, and they only appear once. Thus, m + n - k is equal to the number of numbers that are members of either or both munber. This, by definition, is deg(a ∪ b), completing the proof.

4.3 Supremum and infimum of munbers

DEFINITION. Let a be a munber. Then the supremum of a [sup a] is defined as the least number x such that ∀ numbers a' = a, x ≥ a'.

DEFINITION. Let a be a munber. Then the infimum of a [inf a] is defined as the greatest number x such that ∀ numbers a' = a, x ≤ a'.

As with sets, the supremum and infimum are often referred to as, respectively, the least upper bound and greatest lower bound. If a munber is unbounded above, it has no supremum, and if it is unbounded below, it has no infimum.

THEOREM 4.2. SUPREMUM-INFIMUM THEOREM. Let a be a munber. If a has a supremum, then -a has an infimum, given by inf -a = -sup a.

Proof. For this proof, we continue the tradition of leaving semi-trivial proofs as exercises.

THEOREM 4.3. ADDITIVE PROPERTY. Let a and b be munbers. If they both have suprema, then sup (a + b) = sup a + sup b. If they both have infima, then inf (a + b) = inf a + inf b.

Proof. On page 27 of Apostol's Calculus Volume I, he proves this property for sets, and the proof is much the same for munbers. Notice his definition of C: he unwittingly discovered munbers, but failed to recognize their importance!

EXAMPLE 1. Let a ≡ «2, 6, 7». Then sup a = 7 and inf a = 2. In fact, for any munber of finite degree, sup a is the greatest number equal to a and inf a is the least number equal to a.

EXAMPLE 2. Let b ≡ «1, 1/2, 1/4, 1/8, ...». Then sup b = 1 and inf b = 0, even though b ≠ 0.

EXAMPLE 3. Let c be equal to all integers. Then c has neither supremum nor infimum. Note that in much of this book so far, we have been referring to munbers of finite degree, but the reader may verify that most properties of finite-degree munbers hold for infinite-degree munbers as well.

4.4 Exercises

1. Find the suprema and infima of the following munbers.
   

   (a) «2, 3»	(d) «(n - 1)/n» ∀ n ∈ Z+
   (b) «2»	(e) «sin n» ∀ n ∈ Z+
   (c) «4, 9, 16, 25, 36, ...»	(f) «-5, -4»

2. Prove Theorem 4.2.
3. Suppose that munbers a and b both have suprema and infima. Express the supremum and infimum of a - b in terms of the suprema and infima of a and b.
4. Is the Intersection Degree Theorem valid for infinite-degree munbers? Examine this in two possible cases: that the degree of the intersection is or is not infinite.

4.5 Maximum, minimum of munbers

DEFINITION. Let a be a munber. Then the maximum of a [max a] is the greatest number equal to a, and the minimum of a [min a] is the least number equal to a.

EXAMPLE 4. Let a ≡ «2, 6, 7». Then max a = 7 and min a = 2. As the next theorem indicates, this munber has maximum equal to its supremum and minimum equal to its infimum.

EXAMPLE 5. Let b ≡ «1, 1/2, 1/4, 1/8, ...». Then max b = 1 and min b does not exist. By definition, max b = b = min b.

EXAMPLE 6. Let c be equal to all integers. Then c has no maximum and no minimum.

THEOREM 4.4. Let a be a munber. If a has a maximum, then it has a supremum which is equal to its maximum. If a has a minimum, then it has an infimum which is equal to its minimum.

Proof. This follows directly from the definitions of the associated terms.

These and other functions of munbers are motivated by their use in statistics. Munbers can be quite useful in statistics because of the possibilities of addition and multiplication of data sets.

4.6 Statistical functions of munbers

Some functions which derive from statistics are also useful in other pursuits.

DEFINITION. Let a ≡ «a₁, a₂, ... a_n» be a munber. Then the sum of a [∑ a] is equal to a₁ + a₂ + ... + a_n.

DEFINITION. Let a and b be munbers. Then the quantity a + b is said to be a nonoverlapping sum iff deg(a + b) = deg(a)deg(b).

THEOREM 4.5. NONOVERLAPPING SUM THEOREM. Let a and b be munbers with a nonoverlapping sum. Then ∑(a + b) = deg(b)∑ a + deg(a)∑ b.

Proof. Given that the sum of a and b is nonoverlapping, we know that deg(a + b) = deg(a)deg(b). This implies that «a₁ + b₁, a₁ + b₂, ... a₁ + b_n, a₂ + b₁, a₂ + b₂, ... a₂ + b_n, ... a_m + b₁, a_m + b₂, ... a_m + b_n» has degree mn. More importantly, all of these values are unique, so that ∑(a + b) = (a₁ + b₁) + (a₁ + b₂) + ... + (a₁ + b_n) + (a₂ + b₁) + (a₂ + b₂) + ... + (a₂ + b_n) + ... + (a_m + b₁) + (a_m + b₂) + ... + (a_m + b_n) = n(a₁ + a₂ + ... + a_m) + m(b₁ + b₂ + ... + b_n) = deg(b)∑ a + deg(a)∑ b.

DEFINITION. Let a be a munber. Then the mean of a [E(a)] is equal to (∑ a)/deg(a).

THEOREM 4.6. Let a and b be munbers with a nonoverlapping sum. Then E(a + b) = E(a) + E(b).

Proof. The reader is left with the responsibility of proving this.

Although munbers have not been put to use in statistics, and indeed, they have some shortcomings in that respect, there are several other things to do with them. For example, the median of a munber is defined in the same way as with sets. Therefore, the munber's median is equal to itself if and only if its degree is odd. [The reader may wish to prove this for amusement.] The variance of a munber is also the same, ie, E(a²) - E(a)², so the standard deviation σ(a) = sqrt(var(a)).

4.7 Exercises

1. Find the maxima and minima of the following munbers. Note that the converse of Theorem 4.4 may not be true.
   

   (a) «2, 3»	(d) «(n - 1)/n» ∀ n ∈ Z+
   (b) «2»	(e) «sin n» ∀ n ∈ Z+
   (c) «4, 9, 16, 25, 36, ...»	(f) «-5, -4»

2. Repeat Exercise 1, finding the sum and mean of the given munbers.
3. Repeat Exercise 1 again, finding the variance and standard deviation of the given munbers.
4. For each of the following pairs of munbers, find the degree of their sum. Does the sum overlap? If so, how many times is each number in the sum contributed? In other words, if the sum overlaps, there is more than one way to make some of the values in the sum. How many ways are there to make each possible sum?
   

   (a) «3, 4», «6, 7»	(e) «1, 2, 3», «1, 2, 3»
   (b) «3, 4», «6, 9»	(f) «0, π, 2π», «0, 1»
   (c) «0, 1, 2», «-1, 0»	(g) «-22, -14», «-14, -1»
   (d) «5», «-5, 5»	(h) «-22, -14, -1», «1, 14, 22»

5. Prove Theorem 4.6.
6. TEASER: Use the results from Exercise 4 for this exercise. If the sum does not overlap, verify the results of Theorem 4.5. If the sum does overlap, look at your results for the number of ways to make each sum. Multiply each value by its frequency and sum these. Compare this with the right side of Theorem 4.5. What might this suggest about the importance of the frequency of numbers of a munber?