Videos (~20 minutes)

Reading (~30 minutes)

Warmup (~50 mins)

Problem 1

At the end of this course, every student will receive a letter grade. Consider the relation of the form \((\mathrm{Student}, \mathrm{Grade})\). Here are some example entries of this relation:

\[ \begin{aligned} &(\mathrm{Xenith} ,&\quad B+&) \\ &(\mathrm{Grace} ,&\quad A\phantom{+}&) \\ &(\mathrm{Eun} ,& \quad B-&) \\ &(\mathrm{Natasha} ,&\quad A-&) \\ \vdots \end{aligned} \]

Part A

Explain why this relation is a function. Then, describe the sets that are its domain and codomain.

Part B

Is this function injective, surjective, or bijective? For each, state either “definitely,” “definitely not,” or “it depends.” Explain your answer in each case.

Problem 2

For each of the following situations:

  1. Write a recursive definition for a function \(f\) that describes the specified quantity. Please include the initial condition!!
  2. Carefully describe the domain and codomain of \(f\).

Part A

You place \(C\) dollars into a bank with an interest rate of \(k\%\). This means that your money grows by a factor of \(1 + \frac{k}{100}\) each year. The function \(f(n)\) describes the amount of money in your account after \(n\) years.

Part B

You are running 4 times a week to improve your endurance. You start being able to run 1 mile without stopping. Each week, you are able to run \(\frac{1}{10}\) of a mile longer without stopping. The function \(f(n)\) describes the distance you can run without stopping after week \(n\).

Part C

The factorial function \(f(n) = n!\) computes the product of all integers from \(1\) to \(n\). For example, \(5! = 1 \times 2 \times 3 \times 4 \times 5 = 120\).

Problem 3

Consider the function \(f:\mathbb{N} \rightarrow \mathbb{N}\) given by \(f(1) = 1\) and \(f(n) = f(n-1) + n - 1\).

  1. Compute \(f(4)\) by hand.
  2. Write a recursive function in either Python or Java that implements this function. Use it to compute \(f(4)\) and check that you got the same result.



© Phil Chodrow, 2023