MATH 401/501 - HOMEWORK PROBLEMS - Spring 2025

Numbered exercises are from Tao's Book. For these problems you can only assume results and exercises prior to the given exercise (if you are going to use a previous result, make sure you state it, of course if it has not been proved in class and is not one of the other homework problems, you can attempt to prove those intermediate results, but this is not required.)

Homework 2 (due on Thursday Feb 06, 2025 at 11:59pm):
Knowledge is off. You can use any fact that has been proved before the exercise (including prior exercises in the book), unless the exercise asks you to prove a Theorem/Lemma/Proposition, in that case you can use any fact before the Theorem/Lemma/Proposition. For example, if you are asked to prove item (d) in a proposition, you can use prior items (a), (b), and (c).

  1. Section 2.2: Exercise 2.2.2 (existence of a predecessor for positive natural numbers).
  2. Not in the book: Let a, b, c, d be in N, if a < b and c< d then a+c < b+d (can use anything in Section 2.2 or before).
  3. Section 2.3: Exercise 2.3.2 (N has no zero divisors).
  4. Section 2.3: Exercise 2.3.5 (Euclid's Division Lemma, in prior editions called Euclidean Algorithm, show also uniqueness, not just existence).
  5. Not in the book: use truth tables to prove the distributive property for OR and AND, that is if p, q, r are statements then p or (q and r) is equivalent to (p or q) and (p or r). Also show that no (p and q) is equivalent to no p or no q.
  6. Bonus: Prove Proposition 2.2.12(e): if a< b then a++<=b.
Reading Assignment: Sections 2.3, 3.1 (sets) and 3.3 (functions) (you are more than welcome to read also Section 3.2 (Russel's Paradox) I will not lecture on this however).

Homework 1 (due on Thursday Jan 30, 2025 at 11:59pm)
For the first two problems turn on your number knowledge, for the last two problems turn it off:

  1. Show by induction that for all n natural numbers 0+1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.
  2. Show by induction that n^2 <= 2^n for all natural numbers n>=4. (you will need at some point to show that 2n+1<=2^n for n>=4, or to show that n^2>= 2n+1 for n>=4 depending whether you go right to left or left to right in your inequality, that will be an auxiliary lemma that you may prove by induction! in fact these auxiliary inequalities are true for n>=3).
  3. Show that for all n in N, n++=1+n=n+1 using only the definition of addition not the commutative property. Then show that 1+1=2.
  4. Show that multiplication in N is commutative, that is nxm=mxn for all n,m in N (Exercise 2.3.1 in the book).
    Hint: prove it by induction on n for each m fixed, and you will need two auxiliary lemmas similar to those used when proving commutativity of addition in class.
Reading Assignment Chapter 2.
(Note that a>=b means a is bigger or equal than b, a <= b a is less than or equal to b, for those of you who know LaTeX, respectively \geq and \leq.)

Homework 0 (due on Thursday Jan 23, 2025 at 11:59pm)
This homework you will find in Canvas, is a short exploratory survey.

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: Jan 19, 2023