Practice Problems Week 8

You may collaborate and are encouraged to collaborate with your peers. If you do, be sure to understand the material and write it out in your own words.

There is no requirement to submit the material, but this will help you solidify the concepts.

Problems

problem 1

Consider the function $f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4\}$ represented in the graph above:
1. is f injective? explain
2. is f surjective? explain
At the end of the semester a teacher assigns letter grades to each of her students. Argue that this is a function, and describe its domain and codomain.
Let $S = \{\frac{p}{q}: p, q \in \mathbb Z, q \neq 0\}$ be the set of fractions. We define a relation R on S such by the formula: $\frac{a}{b} R \frac{c}{d} \iff ad = bc$. Is R an equivalence relation? why or why not?
Consider the following function: $f: \mathbb N \to \mathbb N$ described as $f(n+1) = \begin{cases}\frac{f(n)}{2}\quad if\; f(n)\; is\; even \\ 3f(n) + 1 \quad if \; f(n) \; is\; odd \end{cases} $

Notice that if f(0) = 1, we get that: f(1) = 4, f(2) = 2, f(3) = 1, f(4) = 4, etc. So we have a cycle.

if f(0) = 5, can f be injective? Explain why or give a specific example of two elements from the domain with the same image.
if f(0) = 3, can f be injective? Explain why or give a specific example of two elements from the domain with the same image.
show that no matter the initial value, f can not be surjective.