---
tags: mth325-f22
---

# Content Skill Standard Quiz 7

:::info
This quiz contains *new versions* of **Content Skill Standards P.1-P.3, G.1-G.8, ad DR.1** and *introduces* **Content Skill Standards G.8 and DR.2-DR.6**.

**Instructions**

* Work only the problems that you **need to work** and **feel ready to work**.
* Do your work on separate pages with **each Content Skill Standard on its own separate page**. *Please do not put work for multiple Standards on the same page.*
* Make sure to consult the [Specifications for Student Work in MTH 325](https://hackmd.io/lD6oyEN5RdiUi_wdg-rkZg) document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt.
* Also, make sure to carefully read the **Success Criteria** below each problem to know exactly what is expected from that problem.
* When you are done: **Scan** each Learning Target to a clear, legible, black-and-white PDF and **upload the PDF** to the designated folder on Blackboard. Please *do not just take a picture with your camera* --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.

**Authorized resources you may use:**

- [Your textbook](https://discrete.openmathbooks.org/dmoi3.html)
- Previous Content Skill Standard quiz work including feedback on your work
- Any video or reading used in a Daily Prep
- A calculator or computer if used for basic calculations

**The use of Python and networkX is not allowed.**

If you have another resource you'd like to use that it not shown above, please ask before using it.
:::

## Content Skill Standard P.1

:::warning
★ P.1: Given a statement to prove with mathematical induction, I can identify the predicate, state and prove the base case, state the inductive hypothesis, and outline the rest of the proof.
:::

Consider the statement:

>For all positive integers $n$, $1 + 2 + \cdots + n = \dfrac{n(n+1)}{2}$.

1. State the predicate involved in this statement.
2. State and prove the base case for a proof by induction of this statement.
3. State the inductive hypothesis for a proof by induction of this statement.
4. Outline the remainder of a proof by induction for this statement by clearly stating what you would prove, and giving at least one plausible idea for proving it.

**Success criteria:** The predicate must be clearly stated; the base case must be correctly stated and proven; and the inductive hypothesis must be correctly stated in the context of the problem. The outline for the remainder of the proof must clearly state what you are going to prove, and the "plausible idea" must be some specific step you might take that makes sense in the context of the problem.

## Content Skill Standard P.2

:::warning
★ P.2: Given a conditional statement, I can state the assumptions and conclusions for a direct proof, proof by contrapositive, and proof by contradiction.
:::

Consider the statement:

>For all integers $a$ and $b$, if $a$ is odd and $b$ is even, then $a+b$ is odd.

1. Clearly state what you would assume and what you would prove, if proving this statement with a *direct proof*.
2. Clearly state what you would assume and what you would prove, if proving this statement with a *proof by contrapositive*.
3. Clearly state *all* the assumptions you would make, if proving this statement *by contradiction*.

**Success criteria:** All parts of each of the three items here are correctly and clearly stated.

## Content Skill Standard P.3

:::warning
P.3: I can conduct a critical analysis of a proposed proof of a logical proposition.
:::

Below is a proposition and a proposed proof. Read both carefully and then categorize them in one of the following ways:

- *The proposition is false.* If this is the case, give a specific counterexample that shows the proposition is false.
- *The proposition is true but the proof has a fundamental flaw.* If this is the case, find at least one of those fundamental flaws and explain why it makes the proof incorrect.
- *The proposition is true and there are no fundamental flaws in the proof.* If this is the case, say so, and then give one possible suggestion for improving the clarity or correctness of the proof.

Be sure to state which of the three categories to which you believe the proposition and proposed proof belong.

:::success
**Proposition:** For all integers $n$, if $n^2$ is even, then $n$ is even.

**Proposed proof:** We prove this with induction. The base case is $n=0$. We see that $0^2$ is even and $0$ is also even, so the base case holds. Now suppose that for some integer $k$, if $k^2$ is even then $k$ is even. We want to prove that if $(k+1)^2$ is even then $k+1$ is even. We will do this by contradiction. So suppose that $(k+1)^2$ is even. Then expanding this out, we get $k^2 + 2k + 1$. By the induction hypothesis, $k^2$ is even. Therefore $k^2 + 2k$ is even since it is the sum of two even integers. But that means that $k^2 + 2k + 1$ is odd, which is a contradiction because we assumed this number was even. Therefore $k+1$ is even, which is what we wanted to show.
:::

**Success criteria:** The work clearly states which of the three categories the proposition and proof belong to. If there is a counterexample, it is specifically and clearly stated. If there is a fundamental flaw, it is specifically and clearly stated, and the nature of the flaw is clearly explained. If there are no fundamental flaws, this is clearly stated and a plausible suggestion for improvement is given.

## Content Skill Standard G.1

:::warning
★ G.1: I can represent a graph in different ways and change representations from one to another.
:::

Consider the graph $G$, given as the adjacency matrix:

|vertex |0|1|2|3|4|5|6|7|
|---|---|---|---|---|---|---|---|---|
|0|0\.0|0\.0|1\.0|0\.0|1\.0|1\.0|0\.0|1\.0|
|1|0\.0|0\.0|0\.0|1\.0|0\.0|0\.0|1\.0|0\.0|
|2|1\.0|0\.0|0\.0|1\.0|0\.0|1\.0|1\.0|0\.0|
|3|0\.0|1\.0|1\.0|0\.0|0\.0|0\.0|1\.0|1\.0|
|4|1\.0|0\.0|0\.0|0\.0|0\.0|1\.0|0\.0|0\.0|
|5|1\.0|0\.0|1\.0|0\.0|1\.0|0\.0|0\.0|0\.0|
|6|0\.0|1\.0|1\.0|1\.0|0\.0|0\.0|0\.0|0\.0|
|7|1\.0|0\.0|0\.0|1\.0|0\.0|0\.0|0\.0|0\.0|

Write $G$ as:

1. A Python dictionary
3. An edge list

**Success criteria:** All representations are correct, with up to two errors (for example, forgetting one node, or getting one bit wrong in the matrix) allowed.

## Content Skill Standard G.2

:::warning
★ G.2: I can determine information about a graph and its individual vertices and edges using different representations.
:::

Consider the graph $G$, given as the adjacency matrix:

|vertex |0|1|2|3|4|5|6|7|
|---|---|---|---|---|---|---|---|---|
|0|0\.0|0\.0|1\.0|0\.0|1\.0|1\.0|0\.0|1\.0|
|1|0\.0|0\.0|0\.0|1\.0|0\.0|0\.0|1\.0|0\.0|
|2|1\.0|0\.0|0\.0|1\.0|0\.0|1\.0|1\.0|0\.0|
|3|0\.0|1\.0|1\.0|0\.0|0\.0|0\.0|1\.0|1\.0|
|4|1\.0|0\.0|0\.0|0\.0|0\.0|1\.0|0\.0|0\.0|
|5|1\.0|0\.0|1\.0|0\.0|1\.0|0\.0|0\.0|0\.0|
|6|0\.0|1\.0|1\.0|1\.0|0\.0|0\.0|0\.0|0\.0|
|7|1\.0|0\.0|0\.0|1\.0|0\.0|0\.0|0\.0|0\.0|

(a) State the degree sequence of the graph and briefly explain your reasoning **without using a visualization of the graph -- your explanation must only use the edge list**.

(b) Determine if the graph has any cycles of length 3 (triangles). If it does, state one of them as a sequence of vertices and explain your reasoning **without using a visualization of the graph -- your explanation must only use the edge list.**

**Success criteria:** All answers are correct and explanations are complete, correct, and clear (and do not use visualizations of the graph).

## Content Skill Standard G.3

:::warning
G.3: I can give examples of graphs having combinations of various properties or explain why no such example exists, and I can draw examples of special ("named") graphs.
:::

For each situation below, either give an example or explain clearly why no such example exists.

1. A graph with degree sequence 4, 4, 2, 1
2. A graph with 8 vertices and 2 edges
3. A graph with 2 vertices and 8 edges
4. A graph that has a Hamilton path but no Euler path

**Success criteria:** All parts are correct with specific, correct examples given, or in the case of an impossible situation a correct and clear explanation is given.

## Content Skill Standard G.4

:::warning
G.4: I can determine whether two graphs are isomorphic; I can give an explicit isomorphism if they are, and an explanation if they are not.
:::

Are the two graphs below isomorphic?


If you believe they are isomorphic, give a specific isomorphism between them. If you believe they are not isomorphic, explain why not in *specific* terms.

Your explanations must use specific facts about graph isomorphisms. They cannot be general or vague, for example "You can (or can't) rearrange one graph to get the other". [You may use any of the isomorphism invariants found on this list](https://gist.github.com/RobertTalbert/2dadae14f8f42c3654a8a77ef5a038f1) if you like.

**Success criteria:** If a pair of graphs is isomorphic, a correct isomorphism is given as a table of inputs and outputs. If a pair of graphs is *not* isomorphic, a correct reason is given (which can be pulled from [the list of isomorphism invariants](https://gist.github.com/RobertTalbert/2dadae14f8f42c3654a8a77ef5a038f1)).

## Content Skill Standard G.5

:::warning
G.5: I can give a valid vertex coloring for a graph and determine a graph's chromatic number.
:::

Consider the following graph:


Determine the chromatic number of this graph by doing the following:

1. Give a proper vertex coloring (either by labeling the vertices, or by actually coloring them) that uses a number of colors equal to the chromatic number; and
2. Explain why the chromatic number is *exactly* this quantity and not greater or lesser.

**Success criteria:** In part 1, a proper coloring is explicitly stated that uses the stated amount of colors. In part 2, the chromatic number is correct; a proper coloring with the right number of colors is given; and there is an explanation for why a smaller coloring is impossible.

## Content Skill Standard G.6

:::warning
G.6: I can determine whether a graph has an Euler path or Euler circuit, and whether a graph has a Hamiltonian path or cycle.
:::

Consider the graph $G$, given as the adjacency matrix:

|vertex |0|1|2|3|4|5|6|7|
|---|---|---|---|---|---|---|---|---|
|0|0\.0|0\.0|1\.0|0\.0|1\.0|1\.0|0\.0|1\.0|
|1|0\.0|0\.0|0\.0|1\.0|0\.0|0\.0|1\.0|0\.0|
|2|1\.0|0\.0|0\.0|1\.0|0\.0|1\.0|1\.0|0\.0|
|3|0\.0|1\.0|1\.0|0\.0|0\.0|0\.0|1\.0|1\.0|
|4|1\.0|0\.0|0\.0|0\.0|0\.0|1\.0|0\.0|0\.0|
|5|1\.0|0\.0|1\.0|0\.0|1\.0|0\.0|0\.0|0\.0|
|6|0\.0|1\.0|1\.0|1\.0|0\.0|0\.0|0\.0|0\.0|
|7|1\.0|0\.0|0\.0|1\.0|0\.0|0\.0|0\.0|0\.0|

1. Determine if $G$ has an Euler path; if it does, state that path as a sequence of nodes, and if not, clearly explain why not. Then, repeat this task for Euler cycles. *Do these without referring to visual representations of the graph.*

2. Determine if $G$ has a Hamilton path (not necessarily a cycle). If it does, state that path as a sequence of nodes. If not, explain why not. *Do these without referring to visual representations of the graph.*

**Success criteria:** If a graph has a structure (Euler path, Euler cycle, Hamilton path --- whatever is requested in the problem) then a correct example is stated as a node sequence. If not, a correct and *specific* explanation is given. Also, the explanations do not refer to or depend on a visual representation of the graph.

## Content Skill Standard G.7

:::warning
G.7: I can use Prim's Algorithm and Kruskal's Algorithm to construct a minimum spanning tree for a weighted graph.
:::

Consider the weighted graph:


1. Using **Prim's Algorithm starting at node H**, construct a minimum spanning tree for this graph. Your answer here should be a list of edges in the MST, given in the order in which they are added to the tree by Prim's Algorithm.

2. Using **Kruskal's Algorithm**, construct a minimum spanning tree for this graph. Your answer here should be a list of edges in the MST, given in the order in which they are added to the tree by Kruskal's Algorithm.

**Success criteria:** In each part, the minimum spanning tree is given as a correct list of edges, in the correct order of addition into the tree.

## Content Skill Standard G.8

:::warning
G.8: I can use Dijkstra's Algorithm to find a minimum-weight path between two vertices in a connected weighted graph.
:::

Consider the weighted graph:

Starting at node F, use Dijkstra's Algorithm to create the table of shortest-distance information given by the algorithm. Remember the heading for this table looks like this:

| Vertex | Distance from 1 | Previous vertex |
| ----- | ----- | ----- |

**Success criteria:** No more than one error is present in the table. (Errors that "cascade" from a single error count as multiple errors.)

## Content Skill Standard DR.1

:::warning
★ DR.1: I can determine information about a directed graph and its individual vertices and edges using different representations.
:::

Consider the directed graph below:

1. Give the in-degree and out-degree of each vertex.
2. Write the adjacency matrix for the graph.
3. Write the edge list for the graph.

**Success criteria:** Part 1 has no more than one incorrect labeling. Parts 2 and 3 are completely correct.

## :new: Content Skill Standard DR.2

:::warning
DR.2: I can give examples of relations on a set that have combinations of the properties of reflexivity, symmetry, antisymmetry, and transitivity.
:::

For each item below, draw a directed graph for a relation on the set $\{0,1,2,3\}$ that has the indicated combination of properties. If no such example is possible, explain why. (You do not need to provide explanations for examples that are possible.)

1. Symmetric and transitive but not reflexive
2. Antisymmetric and reflexive, but not transitive
3. Reflexive and transitive but not symmetric

**Success criteria:** All examples have the correct combination of properties. Explanations in the event that an example is impossible clearly express why the example is impossible.

## :new: Content Skill Standard DR.3

:::warning
DR.3: I can determine if a relation is an equivalence relation; I can determine the equivalence class of an element under an equivalence relation and determine whether two elements belong to the same equivalence class.
:::

1. Consider the relation on the set $\mathbb{Z}$ of all integers given by saying $a \sim b$ if $a \leq b$. Is this an equivalence relation? If so, give a complete and clear explanation of why. If not, give a *specific* example that explains why not.
2. Consider the relation on the set $\mathbb{N} = \{0, 1, 2, 3, \dots \}$ given by saying $n \sim m$ if $n$ and $m$ have the same number of digits. It can be shown (but don't do it here) that this is an equivalence relation. State five different elements of $[199]$.

**Success criteria:** The explanation in the first item is clear, correct, and specific. The elements listed in the second part are all correct.

## :new: Content Skill Standard DR.4

:::warning
DR.4: I can find the $n$th order composition of a relation with itself.
:::

Consider the relation $r$ on the set $\{0,1,2,3,4,5\}$ given below as an edge list:

[(0, 5), (1, 4), (2, 0), (2, 1), (3, 0), (3, 1)]

1. State the complete edge list for the relation $r^2$.
2. State three edges that belong to $r^3$. (If there are fewer than three, state them all.)

**Success criteria:** The edge list in the first part is correct except for one mistake or omission allowed. The three edges in the second part are all correct members of $r^3$.

## :new: Content Skill Standard DR.5

:::warning
DR.5: I can sketch the transitive closure of a relation as a directed graph.
:::

Consider the relation $r$ on the set $\{0,1,2,3,4\}$ given below as directed graph:


Draw the directed graph for the transitive closure of $r$.

**Success criteria:** The digraph is correct with up to two mistakes or omissions allowed.

## :new: Content Skill Standard DR.6

:::warning
DR.6: I can determine when a set with a relation is a partially ordered set; I can draw the Hasse diagram of a poset and identify maximal/minimal elements and/or greatest/least elements, if they exist.
:::

Consider the set $\{1, 2, 3, 4, 5, 6, 8, 12, 15, 16\}$ and the "divides" relation on that set ($a \sim b$ if $a$ divides $b$). This makes the set a partially ordered set (and you do not need to explain why).

1. Draw the Hasse diagram for this poset.
2. State the maximal and minimal elements, if they exist. If either of these does not exist, say so.
3. State the greatest and least elements, if they exist. If either of these does not exist, say so.

**Success criteria:** The Hasse diagram is correct with no errors or omissions. The maximal, minimal, greatest, and least elements are all correct with no omissions or errors; if any of these does not exist, that fact is clearly stated.