---
tags: 225-spr22, mth225, lt-quiz
---

# Learning Target Final Round

:::info

This quiz contains problems for Learning Targets 5--7 and 9--21.

* It's to your advantage to attempt as many problems as possible. But you *do not* need to attempt all the problems. Only attempt the ones you believe you are ready to take.
* Do your work on separate pages with **each Learning Target on its own separate page**. *Please do not put multiple Learning Targets on the same page.*
* Make sure to consult the [Standards for Assessments in MTH 225](/KoT83ezHRYO3DqPyXMMMag) document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt.
* 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. Remember *do not just take a picture with your camera* --- use a scanning app to create a PDF, and upload the PDF.
:::

:::danger
**All of your work is due on Blackboard by 11:59pm ET on Tuesday, June 21. Please note, the work is not spread over 2 days as in previous Learning Target quizzes. Also no extensions are available as this is part of the final exam.**
:::

## Learning Target 5

:::warning
I can use a truth table to decide if a statement is a tautology or contradiction, or to determine if two statements are logically equivalent.
:::

1. Use truth tables to determine whether the statements $p \wedge (q \vee r)$ and $(p \wedge q) \vee (p \wedge r)$ are logically equivalent or not.
2. Use a truth table to determine whether the statement $(p \rightarrow q) \vee (q \rightarrow p)$ is a tautology.

**Success criteria:** All truth tables have the correct number of rows with no duplicated rows. All intermediate columns are shown. No more than three total errors are permitted. (If you make a mistake in an intermediate column but the rest of the row is correct given that mistake, then the mistake only counts once.) The answers (about whether the statement is a tautology, and about whether the two statements are logically equivalent) are clearly indicated.

## Learning Target 6

:::warning
I can determine the truth value of a predicate at a specific input, and given a quantified predicate I can state its negation and its truth value.
:::

Consider the predicates:

- $P(x)$: $\sqrt{x}$ is an integer
- $Q(x)$: $x=x+1$

The domain of each predicate is the set of all integers (positive, negative, and zero).

1. For each of the following, state whether the expression is **True**, **False**, or **Undetermined**.
(a) $P(100)$
(b) $Q(0)$
(c) $\exists x Q(x)$
(d) $\forall y (P(y))$
2. State the negation of: *Even prime numbers exist.*. Do not simply put "not" or "it is not the case that" in the sentence.

**Success criteria:** All the answers in the first item are correct, and the second item is correct.

## Learning Target 7

:::warning
(**CORE**) I can write a set using roster and set-builder notation.
:::
1. Write each of the following sets in correct roster notation:
(a) $\{n \, \% \, 5 \, : \, x \in \mathbb{N} \}$
(b) $\{ x \in \mathbb{N} \, : \, x^2 \leq 10 \}$

2. Write the following sets in set-builder notation. There may be more than one correct way to do it. Your answer must restate set membership in some way. (*See below*)
(a) $\{2, 4, 8, 16, 32, \dots \}$
(b) $\{1, 5, 9 \}$

3. Consider the sets:
$$\begin{array}
AA &= \{a,b,c\} \\
B &= \{b,c,d\} \\
C &= \{a,c\} \\
D &= \{d,c,b\}
\end{array}
$$
For each statement below, state whether the statement is True or whether it is False.
(a) $\emptyset \subseteq A$
(b) $\emptyset \in A$ (How is this different than the previous question?)
(c) $b \in C$
(d) $a \in D$
(e) $A = B$

**Success criteria:** At least 3 of the 4 answers in the first two items are correct, and no more than two simple errors are present. All answers in the third item are correct.

## Learning Target 9

:::warning
I can determine if a mapping is a function; identify the domain, range, and codomain of a function; and determine the image of a specific input.
:::

Below are three mappings from $\{1,2,3,4\}$ to $\{x,y,z,t\}$. For each one, **state whether the mapping is a function or not a function**. If a mapping is *not* a function, give a specific and clearly-stated explanation. Otherwise, if a mapping is a function, state the domain, codomain, and range (no explanation needed).

1. The mapping $f$ defined by $f(1) = t$, $f(2) = t$, $f(3) = z$
2. The mapping $g$ defined by this table:

| Input | $1$ | $2$ | $3$ | $4$ |
| ------ | ---- | ---- | --- | --- |
| Output | $x$ | $y$ | $z$ | $x$ |

3. The mapping $h$ given by the matrix:

\begin{pmatrix}
1 & 2 & 3 & 4 \\
x & y & z & x
\end{pmatrix}

**Success criteria:** At least two of the three items has a completely correct response, meaning that (1) the mapping is correctly labelled as "function" or "not a function"; (2) if it's not a function, there is a clear and specific explanation; (3) if it is a function, the domain, codomain, and range are all stated correctly.

## Learning Target 10

:::warning
I can determine if a function is injective, surjective, and/or bijective.
:::

Below are three functions with their domains and codomains specified. For each one, classify it as

* Injective but not surjective,
* Surjective but not injective,
* Neither injective nor surjective, or
* Bijective.

If a function *does* have a property you do not have to explain why; but if a function *fails* to have a property, you should explain why in as specific of terms as possible. (Do not just state the definition of the property or give a vague reinterpretation of that definition, but refer to specific elements or other characteristics.)

1. $f: \mathbb{Z} \rightarrow \mathbb{Z}$ given by $f(a) = 2a$.
1. $g: \mathbb{N} \rightarrow \mathbb{N}$ given by $g(a) = a^2 - 1$.
1. The mapping $h$ from the set of all lists of integers (such as `[7,3,-2,10]`) to $\mathbb{Z}$ that takes the list and returns the sum of the elements in the list. For example $h([7,3,-2,10]) = 18$.

**Success criteria:** At least two of the three items has a correct answer that is clearly indicated, accompanied by a correct, clearly written explanation (if the function is not bijective). *Both* the answer and the explanation must be clear, correct, and easy to understand and the explanation must be given in precise terms.

:::info
**About the explanation:** You only need to explain if a function *isn't* injective or *isn't* surjective. If the function is bijective, no explanation is necessary at all (but the answer has to be right).

The explanation needs to be *as specific as possible*. For example you can't just say "There's a collision" if you think a function is not injective. Be specific.
:::

## Learning Target 11

:::warning
I can determine the values of an inverse function and the composition of two functions.
:::

1. Let $f: \{1,2,3,4\} \rightarrow \{x,y,z\}$ be the function given by $f(1) = t, f(2) = t, f(3) = x, f(4) = x$. Determine each of the following:
(a) $f^{-1}(t)$
(b) $f^{-1}(\{x,y\})$

2. Let $u: \mathbb{R} \rightarrow \mathbb{Z}$ be the floor function $u(x) = \lfloor x \rfloor$, let $v: \mathbb{R} \rightarrow \mathbb{R}$ be the function $v(x) = x + 5$, and let $w: \mathbb{Z} \rightarrow \mathbb{Z}$ be the function $w(x) = x \, \% \, 5$. For each of the following, determine if the expression can be computed. If it can be computed, state its value. If it cannot be computed, respond with "DNE" (does not exist).
(a) $(u \circ w)(11.2)$
(b) $(w \circ v)(11.2)$
(c) $(w \circ u)(11.2)$

**Success criteria:** At least four of the five answers are correct.

## Learning Target 12

:::warning
I can compute values of the floor, ceiling, factorial, and "mod" functions as well as integer division (`DIV` or `//`).
:::

State the values of the following. \textbf{Give a brief explanation for each of these.}

1. $\lfloor 0.4 \rfloor$
1. $\lfloor -1.4 \rfloor$
1. $\lceil 19.01 \rceil$
1. $\lceil -19.01 \rceil$
1. $9!$
1. $1!$
1. $DIV(467,2)$
2. $467 // 6$
3. $467 \ \% \ 2$
4. $(-19) \ \% \ 5$

**Success criteria:** At least 9 out of 10 answers are correct.

## Learning Target 13

:::warning
(**CORE**) I can apply the Additive and Multiplicative Principles and the Principle of Inclusion/Exclusion to formulate and solve basic combinatorics problems.
:::

Work out the answer to each of the counting questions below. State the answer clearly, and show all work or explain your reasoning in words on each. **Note:** Answers with insufficient explanations or work are not accepted.

1. How many strings of of lowercase letters ("a" through "z") and having length exactly 4 are there, that either start or end with a vowel (the letters "a", "e", "i", "o", or "u")?
1. How many strings are there of lowercase letters ("a" through "z") that have length 4 or less, not counting the empty string?

## Learning Target 14

:::warning
(**CORE**) I can compute a binomial coefficient and apply the binomial coefficient to solve basic combinatorics problems.
:::

1. Compute the exact numerical value of the following. Either show work or explain in one sentence how you got your answer. Leave no fractions in your answers.
(a) $\binom{14}{8}$
(b) $\binom{22}{10}$

2. Solve the following counting problems. State your answer clearly and justify each answer with math work or a brief verbal explanation. The justification for each answer must involve the binomial coefficient.
(a) If you flip a fair coin 10 times and each flip comes up either heads or tails, how many possible outcomes are there that contain exactly two heads?
(b) If you flip a fair coin 10 times and each flip comes up either heads or tails, how many possible outcomes are there that contain at most three tails?

**Success criteria:** All of the answers on the first item are correct and have sufficient justifications. Both parts of the second item have good-faith efforts at a correct answer and contain clear, substantive explanations that justify the approach and answer. At least one of the two answers must be correct and sufficiently justified; the other may be incorrect as long as it results from a "simple" error.

## Learning Target 15

:::warning
I can determine the number of permutations of a set of objects and the number of $k$-permutations from a set of $n$ objects.
:::

Solve each problem below. Show your work or explain your reasoning on each of these, and clearly indicate the numerical answer on each (don't just plug into a formula and leave it).

1. A club has 35 members. How many ways are there to choose four members to serve in the positions of president, secretary, and treasurer? (Assume no person can hold more than once office.)
2. In how many different orders can five runners finish a race if no ties are allowed?

**Success criteria:** Both parts have good-faith efforts at a correct answer and contain clear, substantive explanations that justify the approach and answer. At least one of the two answers must be correct and sufficiently justified; the other may be incorrect as long as it results from a "simple" error.

## Learning Target 16

:::warning
I can use the "dots and dividers" method to count the number of ways to distribute objects among a group.
:::

Solve each problem below. Show your work or explain your reasoning on each of these, and clearly indicate the numerical answer on each (don't just plug into a formula and leave it).

1. You have eight identical copies of a book in your apartment and you want to give away all eight copies to your three best friends. It's OK for one or two of the friends to not get any of the books. How many ways are there to give the books away?
1. Repeat the first problem if you want each friend to get at least one copy.

**Success criteria:** Both parts have good-faith efforts at a correct answer and contain clear, substantive explanations that justify the approach and answer. At least one of the two answers must be correct and sufficiently justified; the other may be incorrect as long as it results from a "simple" error.

## Learning Target 17

:::warning
(**CORE**) I can generate several values of a sequence given either a closed-form or recursive definition.
:::

List the first six (6) terms of each of the following sequences. You do not need to show your work, but your answers must be correct.

1. $a_n = 2(3^n) + 3, \ \text{where} \ n = 1, 2, 3, \dots$
1. $b_n = 3 - 2n \ \text{where} \ n = 0, 1, 2, \dots$
1. $c_0 = 2, \ \text{and} \ c_n = 2c_{n-1} + 3n \ \text{if} \ n > 0$
1. $d_0 = 1, d_1 = 2 \ \text{and} \ d_n = 7d_{n-1} - 10d_{n-2} \ \text{if} \ n > 1$

**Success criteria:** At least three of the four sequences have all six terms correctly listed.

## Learning Target 18

:::warning
I can use "sigma" and "pi" notation to find the sum and product of a sequence of numbers.
:::

1. Compute the numerical value of each of the following:
(a) $\displaystyle{\sum_{n = 1}^4 \frac{n^2(n-1)^2}{3}}$
(b) $\displaystyle{\prod_{n = 2}^5 (n/10)}$
(c) $\displaystyle{\sum_{n = 1}^{10} \left \lfloor \frac{n}{2} \right \rfloor}$ where $\lfloor x \rfloor$ is the floor function

2. For each sum below, write the sum correctly using sigma notation:
(a) $3 + 6 + 12 + 24 + 48$
(b) $1 + 2 + 4 + 8 + \cdots + 2048$

**Success criteria:** At least four of the five items has a correct answer. Up to two simple errors are allowed.

## Learning Target 19

:::warning
I can find closed form and recursive definitions for arithmetic and geometric sequences.
:::

For each sequence below, find both a closed formula and a complete recursive definition for the sequence. You do not need to show your work, but your results must be correct.

1. $1, 2, 4, 8, 16, 32, \dots$
1. $5, 7, 9, 11, 13, \dots$
1. $1, 9, 17, 25, 33, \dots$
1. $0.5, 0.05, 0.005, 0.0005, \dots$

**Success criteria:** At least three of the four items has a complete and correct answer. (Remember each item asks for *two* things.)

## Learning Target 20

:::warning
I can solve a second-order linear homogeneous recurrence relation using the characteristic root method.
:::

Solve the following recurrence relation using the characteristic root method:

$$a_0 = 3, a_1 = 10; \ \text{and} \ a_n = 5a_{n-1} - 6a_{n-2} \ \text{if} \ n > 1$$

**Success criteria:** A correct and complete function is given, and all significant work is shown clearly. Omission of significant algebra or arithmetic steps will constitute an unsuccessful attempt. ==In particular, the steps for solving the characteristic equation and for finding the coefficients of the solution must be fully present.== Up to two (2) simple errors are allowed.

## Learning Target 21

:::warning
(**CORE**) Given a statement to be proven by mathematical induction, I can identify the predicate, prove the base case, state the inductive hypothesis, and sketch a proof.
:::

Consider the statement: **For any positive integer $n$,**
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^n} < 1.$$

Set up the framework for a proof by mathematical induction by doing the following steps:

1. State the base case, and prove that the base case is true.
2. Clearly state the inductive hypothesis.
3. Clearly state the inductive step (that is, clearly state what needs to be proven following the inductive hypothesis).
4. Give at least one suggestion for how a full proof of the inductive step might go, that is reasonable and likely to be useful.

**Success criteria:** The base case is clearly stated and correctly proven. The inductive hypothesis is clearly and correctly stated. The inductive step is clearly and correctly stated.