Problem Set 12—Advanced Function Concepts

Purpose

This problem set reinforces your ability to reason in relatively advanced ways about functions. In particular, it develops your understanding of function compositions, inverses, and how functions interact with set operations.

Background

This exercise is based on sections 6.4 through 6.6 of our textbook. We discussed, or will discuss, this material in class on November 18, November 21, and November 28.

Problem 4 continues the project of challenging you with Prof. Nicodemi’s real analysis problems, and so does not draw on the material we are discussing now in class.

Activity

Solve the following problems. Formal proofs should be word-processed (i.e., not hand-written) and should follow the guidelines in Sundstrom’s text.

Problem 1

Exercise 8a in section 6.4 of our textbook (if f(x) = x + 1, find an expression for f ⁿ(x) and use induction to prove it correct; see the textbook for a definition of the f ⁿ notation).

Problem 2

Exercise 9 in section 6.5 of our textbook (prove that if f : A → B is a bijection, then f ^-1 : B → A is also a bijection).

Problem 3

Exercise 7 from section 6.6 of our textbook (prove that if f : S → T is a function and C ⊆ T, then f(f ^-1(C)) ⊆ C).

Problem 4

Take the following as given:

Fact 1. For any real numbers x > 0 and y, there is a natural number n such that nx > y.

Fact 2. For any real number x there is an integer m such that x ≤ m but x ≰ m - 1.

Now prove the following, but first figure out what they mean by finding examples. Note that the propositions are numbered to continue the numbering in Problem 4 of Problem Set 11:

Proposition 2. For any real numbers x and y such that x < y, there is a natural number n such that ny - nx > 1.

Proposition 3. If x and y are real numbers such that x < y and n is a natural number such that ny - nx > 1, then there is an integer q such that nx < q < ny. Hint: first find m as in Fact 2 for ny and then subtract 1.

Theorem. For any real numbers x and y such that x < y, there is a rational number s such that x < s < y. Hint: this is only one step from Proposition 3.

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.