Department of Mathematics and Statistics
Binghamton University

Math 330: Number Systems
Announcements

Section 6, Zaslavsky · Spring 2024

Contents

• Extra Propositions
• Extra Problems
• Test Grade Guidelines

Extra Propositions

• Proposition 1.28. −(m − n) = n − m. (We would have appreciated having this in §1.3.)
• Proposition 1.17.1. k | n if and only if (−k) | n.
• Proposition 2.3.1. −1 ∉ N.
• Proposition 2.18.1. Prove the following propositions for all natural numbers n.
  1. The sum 1+2+3+...+n = n(n+1)/2, by induction.
  2. 2 | n(n+1), directly from part (i).
  3. The sum 1²+2²+3²+...+n² = n(n+1)(2n+1)/6, by induction.
  4. 6 | n(n+1)(2n+1), directly from part (iii).
  5. 2ⁿ > n, by induction.
• Proposition 2.32.1. Every nonempty subset A ⊆ N has a unique smallest element.

Extra Problems

1. Implications.
   1. P and Q are statements. What is the difference among "P if Q", "P is necessary for Q", and "P only if Q"?
   2. Write the inverse, converse, and contrapositive of the following statement: "There can't be rain if there is no water."
   3. Consider the statement "All apples are pears." Formulate it as an implication P ⇒ Q (say what P and Q are). Then state the contrapositive, converse, and inverse of your implication. Which of the four statements are true and which are false? Explain why.
   4. This original question was a mistake: Consider the statement "No apples are pears." Formulate it as the negation of an implication R ⇒ S (say what R and S are). Then state the contrapositive, converse, and inverse of your implication. Simplify the negations of the original implication and the contrapositive, converse, and inverse.
      Corrected question: Consider the statement "No apples are pears." Formulate it as an implication R ⇒ S (say what R and S are). Then state the contrapositive, converse, and inverse of your implication.
   5. Compare the implications of parts c and d. Are they different? Why, or why not?
   6. Consider the statement "Every natural number is positive." Do the same as part (c) for this statement.


2. Problem 4.3.1 (the "x+2 Problem"). This is just like the 3x+1 and x+1 problems except that x_n+1 = x_n + 2 for odd x_n. Question: What is the long-term behavior of this sequence? How does it depend on the initial number m = x₁?


3. Sets.
   Project 5-A. Among the four sets A, B, F (Project 5.3) and
           H = {3x−21: x ∈ Z and x > 7},
   find all pairs such that (a) X = Y, (b) X ⊆ Y.
   Advice: Check every pair of sets. Prove all your answers thoroughly, using the set definitions. That includes proving non-subsets.


4. Sets: 5.5.
   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 on set definition below.


5. Sets: 5.5+.
   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}.
   
   
   1. 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.
      
      
   2. Now, find the simplest possible way of writing each of the sets in the preceding list.


6. Functions. See Chapter 5 on the advice page for definitions that are not in §5.4.
   1. A certain function g: X → Y is defined by a table of values as follows: X = {cat, squirrel, sunflower, grass}, Y = {0, 1, 2, 3, 4, 5}, and

      x	g(x)
      cat	3
      squirrel	0
      sunflower	1
      grass	4

      
      Here are some questions about g.
      1. What is the domain, Dom(g)?
      2. What is the codomain, Codom(g)?
      3. What is the image, Im(g)?
      4. Is g injective (one-to-one)?
      5. Is g surjective (onto)?
      6. Is g bijective (a one-to-one correspondence)?
      
      
   2. Another function h: X → Y is defined by a table of values as follows, with the same X and Y, and

      x	h(x)
      cat	3
      squirrel	0
      sunflower	2
      grass	0

      
      Here are some questions about h.
      1. What is the domain, Dom(h)?
      2. What is the codomain, Codom(h)?
      3. What is the image, Im(h)?
      4. Is h injective (one-to-one)?
      5. Is h surjective (onto)?
      6. Is h bijective (a one-to-one correspondence)?

Tests

 	A	B	C	D	F

	A	B	C	D	F