Department of Mathematical Sciences
Binghamton University

Math 330: Number Systems
Announcements

Section 5, Zaslavsky · Fall 2017

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Tests and Project
Mathematical Language
Additions and Corrections to the Textbook
for Chapter 1 2 3 4 5 6 8 9 10 11 12 13

Test Coverage; Test and Project Grade Guidelines

Midterm Test

The midterm covers Chapters 1 - 6.

The format is open-book: you can have your textbook, but no notes. Grading guidelines:

 	A	B	C	D	F
	80-100	63-79	46-62	40-45	0-39

Each problem is worth 20 points. The formula for your total score is

score = max( S, (9×S − 5×Q₁)/8 ),

rounded to the nearest integer, where S = sum of all questions and Q₁ = score on Question 1.

Final Exam

The final exam covers the entire course, with some emphasis on the second half (Chapters 8 - 13, excluding 12).

The format is open-book: you can have your physical textbook, but no notes and no electronic devices.

Mathematical Language

In math, "unique" has a very precise meaning: "only one". It never means "different". If you mean "different", say "different", or "distinct" to mean "different from each other.
We often use "unique" as part of a phrase like "There is a unique ...". That means "There is one and only one ...".

"Therefore" and non-synonyms.
- "So" means "therefore". Throwing in random "so"s here and there will confuse your reader and get a lower grade.
- If you do mean "therefore", don't say "and". "And" does not mean "therefore". Good synonyms for "therefore" are "thus", "consequently", and so on.
- The symbol ⇒ means "implies". The symbol ∴ means "therefore". They don't mean the same thing.
  "A ⇒ B" means "If A is true, then B is true." It doesn't assert that A is true and it doesn't assert that B is true.
  "A, therefore B", also written "A ∴ B", means "A is true, and therefore B is true." It does assert that A is true and it also asserts that B is true.
"Assume" means "suppose for this discussion". It does not mean "I'm going to prove" or "This is a known fact".

Pay attention to the difference between a subset and an element of a set. Can you explain the difference? If not, think about it carefully.
Pay attention also to the difference between the notations for a subset of a set, ⊆, and for an element of the set, ∈. (These symbols may not show correctly in all browsers.)

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Additions and Corrections to the Textbook

Chapter 1 2 3 4 5 6 8 9 10 11 13

P AND P is logically equivalent to P.
P OR P is logically equivalent to P.
P AND Q is logically equivalent to Q AND P.
P OR Q is logically equivalent to Q OR P.
(P AND Q) OR R is logically equivalent to (P OR R) AND (Q OR R).
(P OR Q) AND R is logically equivalent to (P AND R) OR (Q AND R).

Proving an "OR" statement. Suppose you have two statements, P and Q. How can you prove the compound statement "P OR Q"? (You also have some assumptions given to you, which you're supposed to use in the proof.) There are (at least) two usual methods.
Method 1. Go through your deduction, using the given assumptions, and at some points you'll have two alternative cases, in one of which you prove P is true, and in the other one of which you prove Q is true.
Method 2. Assume P is false; add this hypothesis to the given assumptions. Then prove Q is true. (You could reverse the roles of P and Q.)

Proving an "AND" statement is much simpler. To prove "P AND Q" you just prove P and also prove Q. That's two separate proofs.

Chapter 4

Example 4.2, the "3x+1 problem", is very mysterious. It shows how much more harder it can be to understand a sequence with a recursive definition than a sequence with a simple formula for each term as in example 4.1.

In Proposition 4.16 the sequences should be (x_j)_j=a^∞ and (y_j)_j=a^∞.

In Proposition 4.17 it should be stated that a, a+r ≥ 1. (If not, then the summations are not both defined. Think about why not.)

In Project 4.26 it should be stated that m, n ∈ Z.

Chapter 5

Pay attention to the difference between an element of a set and a subset of a set.
Suppose you have a set S. A subset of S is a set, and it is not (except in very unusual cases) one of the things that is in the set S. An element or member of S is one of the things that is in the set S; it is not necessarily a set, although it can be (since sometimes the elements of S might be sets themselves).
Pay attention also to the difference betweeen the notations for a subset of S, ⊆ (as in T ⊆ S), and for an element of S, ∈ (as x ∈ S).

When reading or writing a set definition, pay attention to what is a variable inside the set definition and what is not a variable. As examples, how do the following pairs of sets differ? First, we define some sets:
- R = {my: y ∈ Z, m ∈ N, and my > 0}.
- S = {n: n ∈ N}.
- T = {n}.
- U_m = {my: y ∈ Z and my > 0}.
- V = {2y: y ∈ Z and 2y > 0}.
- W_m = {my: y ∈ Z and y > 0}.
Now we state the pairs of sets to think about:
1. S, and T where n ∈ N.
2. R, and U_m where m ∈ N.
3. U_m and W_m, where m ∈ Z.
4. W₂, and W_m with m = 2.
5. V, and U_m with m = 2.
Now, find the simplest possible way of writing each of the sets in the preceding list.
(The book used the previous version of these examples as the basis for
Project 5.5.)

In comparing sets X and Y for equality or containment, there is usually more than one method that may help.
1. There are (at least) two methods for showing two sets equal.
  1. The two-way comparison method described in the "Template for proving A = B" on page 49.
  2. Transform the defining equations or properties of X until you get the description of Y.
2. There are two ways to show two sets are not equal.
  1. Find an element in one that isn't in the other.
  2. Restate them both in comparable ways (as in A2) that make the answer obvious.
3. Two ways to find the intersection of two sets:
  1. Combine the defining properties and (as in A2) simplify.
  2. Show the intersection is empty by proving that no element can be in both sets. (Of course, only if the intersection really is empty.)

In Project 5.5(iii), do you think m is supposed to be the same m for both V_m and W_m? What is the range of m supposed to be?
The problem is not stated clearly enough. (It's my fault; I wrote the problem for them.) The sets should be defined before the subproblem (i, ii, iii) statements. The range of m should appear within each subproblem statement. For instance, (iii) should read:
"V_m and W_m for a specified m ∈ Z."
See the discussion above on "set definition".

Chapter 6

The definition of a partition on page 57 is missing an essential condition: No subset P ∈ Π can be empty.

I like to define the degree of the zero polynomial as −1. Then I don't need to treat it as a special case in Proposition 6.18. Either my way or the book's way is acceptable.

Proposition 6.18 is fine if the coefficients are real numbers or rational numbers. It is also fine if all coefficients are integers and the divisor n(x) is monic. If the coefficients are integers and n(x) is not monic, q(x) and r(x) do not exist with the stated properties. We'll assume the polynomials in the Division Algorithm for Polynomials have coefficients that are real or, if they are integral, the polynomials are monic.

A root of a polynomial is also often called a zero of the polynomial, because it's a value of x that makes the polynomial take on the value 0.

Here is an important extra proposition (that I proved in class).

Proposition 6.23'. Let n ∈ N and m, k ∈ Z. Write m = qn + r and k = pn + s using the division algorithm, so 0 ≤ r, s < n. Then m is congruent to k (mod n) if and only if r = s.
(This proves that my equivalence relation =_n is the same relation as their equivalence relation = (mod n).)

Here are propositions derived from Prop. 6.18 that I discussed in class.

Proposition 6.19A. If you divide p(x) by x−a, the remainder is p(a).

Proposition 6.19. (Alternate statement.) p(x) is divisible by x−a if and only if p(a) = 0.

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Chapter 8

On pages 80, line 4 up, and 81, line 1, Proposition 8.32(ii) should be 8.32(iii) in two places.

For Sect. 8.3, an instructive Example. Let
S = { x in R : x is a rational number and x ≥ √2 }
and let
T = { x in R : x is a rational number and x ≤ √2 }.
Then S has many lower bounds (think of one), but no infimum (greatest lower bound), in the rational numbers. Similarly, T has many upper bounds, but no supremum (least upper bound), in the rational numbers. The reason is that √2 is not a rational number (see the facts, below).
In the real numbers, the facts change! In the real numbers, √2 = inf(S), i.e., it is the greatest lower bound of S in the real numbers. And, √2 = sup(T), i.e., it is the least upper bound of T in the real numbers.
The purpose of this example is to show that Axioms 8.1 and 8.9 are not enough to specify the real numbers (since they're satisfied also by the rational numbers—and other systems). We need the Completeness Axiom, Axiom 8.32, to narrow down to the real numbers.
Facts used in the preceding example, but not proved at this time (they would take us off the subject).

A rational number is a number that is the ratio (i.e., quotient) of two integers. (This is getting ahead of the logical development, but the rational numbers make a good example, so I'll just postpone all the proofs about them till later.)
√2 is not a rational number.

Proposition 8.50 should be stated differently: Proposition 8.50'. If A ⊆ B, and if sup(A) and sup(B) exist, then sup(A) ≤ sup(B).

Chapter 9

Example 9.5: See Section 5.4 for the definition of the graph of f.

In Proposition 9.10(i), we need to assume A is not empty.

There is one property of Z that hasn't been proved of e(Z) (what the book calls "Z as a subset of R"). That is the following:

Corollary 9.21. The Induction Axiom 2.15 is true for Z and N as subsets of R.

(Stated more precisely: it is true for e(Z) and e(N), which are actually subsets of R.)

Chapter 10

The proof of Theorem 10.1 could be simpler. We deduce the existence of a least upper bound u, as in the book. Now, u−1 is not an upper bound because u−1 < u, the least upper bound. Therefore, there exists a natural number n such that u−1 < n. Consequently, u < n+1. But then u is not an upper bound for N, which contradicts the definition of u. Therefore, the assumption that N has an upper bound in R must be false.

What the book calls an increasing sequence (page 101) is usually called "weakly increasing" (or "non-decreasing"), while a sequence such that x_k+1 > x_k for all k is called "strictly increasing" (or "increasing").
Similar terminology ("weakly" or "strictly") is used for decreasing sequences.

There should be a Proposition 10.18&12;. If (x_k) is an increasing sequence, then i < j implies x_i ≤ x_j.
We haven't proved it, but it's so useful in proofs like 10.19 that we should use it (as the book does, without comment, in proving 10.19).

The proof of Theorem 10.25 (page 104) is a bit sloppy (not seriously so) in spots. Here is how I would modify it:

P. 104, line 5: "Thus every element in A is bounded above by 2." This has been proved only for x ∈ A such that x ≥ 1. We should also prove it for x < 1. Can you find an easy proof?
In (a), line 3, "the distance between (u−h)² and u²" is not u² − (u−h)², it is |u² − (u−h)²|. Thus, to prove the distance is < δ, they ought to prove u² − (u−h)² > 0. The book is assuming we know that. In fact, a few lines later at (a), lines 7-8, they prove u−h < u, which we can use to prove (u−h)² < u². (Can you provide that proof?) If they had proved it sooner, they would have fixed this oversight.
For (a), line 5, they again need the fact that (u−h)² < u².
In (a), last line, the reason we have a contradiction could be stated more precisely as "u−h < u and u−h is an upper bound for A, ...". The fact that h > 0 is not the exact reason; it's that u−h is an upper bound which is < u. (Of course you realized what they meant.)
P. 104, line 4 up: "By a similar argument as in" is bad English. Good English would be "By an argument similar to that in".

Chapter 11

Proposition 11.2 should be an if-and-only-if statement: Proposition 11.2 (completed). x/y = z/w if and only if xw = zy.

Project 11.26 should begin: "Use Proposition 11.25 to prove that ..."

Chapter 13

Theorem 13.4 should be accompanied by a corollary: Theorem 13.4'. If a set X has the cardinality of [n] and also the cardinality of [m], where n, m ∈ Z_≥0, then m = n. (I think this helps to explain the meaning of Theorem 13.4.)

Proposition 13.13 has the corollary (which I think is the main result):
Proposition 13.13'. Z is countably infinite.
(Why is it not finite? There are proofs (1) using Proposition 13.6 and (2) using a bijection N → Z.)

The definition of an algebraic number on p. 125 should read "... of a nonzero polynomial ...".

Main class page | Schedule and homework | Announcements | Term Project | Syllabus

Math 330: Number SystemsAnnouncements

Section 5, Zaslavsky · Fall 2017

Contents

Midterm Test

Final Exam

Math 330: Number Systems
Announcements