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

# Learning Target Quiz 6

:::info

This quiz contains questions for Learning Targets 1 through 20.
:::

:::danger
:fire: **This is the first of the *final three main Learning Target quizzes* where all past Learning Targets are available**. Your primary focus on this one needs to be making a successful attempt on any CORE Learning Targets you have not yet completed. Those targets are **Learning Targets 2, 3, 7, 8, 13, 14, and 17**. (Learning Target 21 is also Core and you work with that next week.)

So make sure you focus your efforts on any of those that are not yet completed, and be sure to check your work against the success criteria before submitting!
:::

:::info
* 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.

**Deadlines for your work:**

* Work on *Learning Targets 1-10* is due by **11:59pm ET Thursday, June 9**.
* Work on *Learning Targets 11-20* is due by **11:59pm ET Friday, June 10**.

:::

## Learning Target 1

:::warning
I can represent an integer in base 2, 8, 10, and 16 including negative integers in base 2.
:::

Do **all** of the following:

1. Convert the base 10 integer $99$ to binary. *Show your work and circle your answer*.
2. Convert the base 16 integer $23E$ to decimal. *Show your work and circle your answer*.
3. Convert the base 2 integer `11011010` to octal. *Show your work and circle your answer*.
4. The 8-bit binary representation of the decimal number $99$ is `01100011`. Write the 8-bit binary representation of $-99$ using two's complement notation. *Show your work and circle your answer*.

**Success criteria:** All four answers are correct, and the work leading to each answer is clear and legible. Up to two simple errors are allowed.

## Learning Target 2

:::warning
(**CORE**) I can add, subtract, multiply, and divide numbers in base 2.
:::

Do **all** of the following:

1. Add the base-2 integers `11110001` and `10010011`. *Show your work and circle your answer*.
2. Subtract the base-2 integers `11110001` and `10010011`. *Show your work and circle your answer*.
3. Multiply the base-2 integers `1001` and `11`. *Show your work and circle your answer*.
4. Divide the base-2 integer `10010011` by `11`. *Show your work and circle your answer*.

**Success criteria:** All four answers are correct, and the work leading to each answer is clear and legible. Up to two simple errors are allowed.

## Learning Target 3

:::warning
(**CORE**) Given a conditional statement, I can state its hypothesis, conclusion, negation, converse, inverse, and contrapositive.
:::

Consider the conditional statement: *If I water my plants, then they will grow.*

1. State the **hypothesis** of this statement.
2. State the **conclusion** of this statement.
3. State the **converse** of this statement.
4. State the **inverse** of this statement.
5. State the **contrapositive** of this statement.
6. State the **negation** of this statement (without simply putting "Not" or "It is not the case that" in the front of the statement).

**Success criteria:** All answers are given in clear and correct English (*not* in symbolic notation). The negation and contrapositive must be correct, and no more than one error is allowed in the others.

## Learning Target 4

:::warning
I can construct a truth table for propositions involving 2, 3, or 4 statements.
:::

Construct a correct truth table for each of the following statements.

1. $p \wedge ((\neg p) \vee q)$
2. $(\neg p) \rightarrow (q \vee r)$

**Success criteria:** Both 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.)

## 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 $\neg (p \vee q)$ and $(\neg p) \wedge (\neg q)$ are logically equivalent or not.
2. Use a truth table to determine whether the statement $(p \wedge q) \rightarrow (p\vee q)$ 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:

- $P(x): x-1 \, \text{is even}$
- $Q(x): (x+1) \, \% \, 5 = 0$

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) $P(8)$
(b) $Q(29)$
(c) $\forall x P(x)$
(d) $\forall y (\neg P(y))$
2. State the negation of the statement, "It will rain every day this week" without merely putting the word "not", "It is not the case that", etc. on the statement.

**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) $\{ a \in \{1,2,3,\dots, 10\} \ : \ a/2 \ \text{is an integer} \}$
(b) $\{ \lfloor x/3 \rfloor \, : \, x \in \{1,2,3,4,5,6\}\}$ where $\lfloor x \rfloor$ is the "floor" function

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) $\{2, 5, 8, 11, 14, \cdots \}$
(b) $\{2, 1, 1/2 \}$

3. For each statement below, state whether the statement is True or whether it is False.
(a) $\mathbb{Z} \subseteq \mathbb{N}$
(b) $\emptyset \subseteq \{1,2,3,4,5\}$
(c) $-3 \in \mathbb{N}$
(d) $\{0,1,2,3,4,5\} = \{ x \, \% \, 6 \, : \, x \in \mathbb{N}\}$

**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 8

:::warning
(**CORE**) I can find the cardinality, power set, and complement of a set; and I can find the intersection, union, difference, and Cartesian product of two sets.
:::

Let $X = \{a,c,e,g\}$, $Y = \{d,e,f,g\}$, and $Z = \{x,y,z\}$. The universal set for these is $U = \{a,b,c,d,\dots,y,z\}$. Find each of the following. If the answer is a set, write it in roster notation.

1. $X \cup Y$
1. $Z \cap Y$
1. $Y \setminus X$
1. $(X \cup Z) \cap Y$
1. $Z \times Z$
2. $|{\cal{P}}(X)|$

**Success criteria:** At least five of the six items have correct responses, and no more than two simple errors are present.

## 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 $f(1) = t$, $f(2) = z$, $f(2) = t$, $f(3) = z$, $f(4) = y$
2. The mapping $g$ defined by this table:

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

3. The mapping $h$ given diagram:

**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 $f(a) = a^2$.
1. $g: \mathbb{N} \rightarrow \mathbb{N}$ given by $g(a) = a \, \% \, 8$.
1. The mapping $h$ from the set of all possible 8-bit binary strings, to the natural numbers that maps a string to its weight (that is, the number of 1 bits it contains). For example $h(11010110) = 5$.

**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 $f(1) = y, f(2) = y, f(3) = x, f(4) = z$. Determine each of the following:
(a) $f^{-1}(y)$
(b) $f^{-1}(\{x,z\})$

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 $w(x) = x \, \% \, 10$. 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) $(w \circ u)(18.3)$
(b) $(u \circ v)(11.7)$
(c) $(w \circ v)(11.7)$

**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. $\lfloor 11.7 \rfloor$
1. $\lfloor -10.2 \rfloor$
1. $\lceil 199.1 \rceil$
1. $\lceil -105.9 \rceil$
1. $8!$
1. $0!$
1. $DIV(19999,444)$
2. $444 // 4$
3. $19999 \ \% \ 444$
4. $444 \ \% \ 41$

**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 five-digit hexadecimal (base 16) numbers are there?
1. How many 8-bit binary strings are there that have either a `1` in the leftmost bit, or a `11` in the rightmost two bits?

## 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) $\binom{20}{12}$
(b) $\binom{20}{0}$
(c) $\binom{20}{1}$

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) How many 8-bit binary strings have exactly 6 `1` bits?
(b) How many 8-bit binary strings have *at least* 6 `1` bits?

**Success criteria:** All three 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. How many rearrangements of the word `LEARN` are there?
1. Suppose you are playing a word game where you have to make as many four-letter words as possible given a set of ten letters which are all different. How many such words can be made? (For this problem, the "words" don't have to be actual English words; for example `FPCK` is an acceptable four-letter word.)

**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. How many natural number solutions are there to the equation $x + y + z + t = 100$?
2. How many natural number solutions are there to the equation $x + y + z + t = 100$ if all the variables have to be positive (not just natural numbers), $x$ must be greater than or equal to $5$, and $t$ must be greater than or equal to $10$?

**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. $a_n = 3(2^n) -1 , \ \text{where} \ n = 1, 2, 3, \dots$
1. $b_n = 4 + \frac{n}{3} \ \text{where} \ n = 0, 1, 2, \dots$
1. $c_0 = 2, \ \text{and} \ c_n = 5c_{n-1} + n \ \text{if} \ n > 0$
1. $d_0 = 1, d_1 = 2 \ \text{and} \ d_n = d_{n-1} - 2d_{n-2} \ \text{if} \ n > 1$

**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) $\displaystyle{\sum_{n = 1}^4 (3(2)^n - 1)}$
(b) $\displaystyle{\prod_{n = 2}^4 (1 + 2n)}$
(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) $1+ 4 + 7 + 10 + 13$
(b) $2 + 4 + 8 + 16 + 32 + \cdots + 1024$

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

## :new: 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. $1, 5, 9, 13, 17, \dots$
1. $2, 4, 8, 16, 32, \dots$
1. $2, 4, 6, 8, 10, 12, \dots$
1. $1, 0.1, 0.01, 0.001, 0.0001, \dots$

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

## :new: 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:

$$a_0 = 1, a_1 = 5; \ \text{and} \ a_n = 6a_{n-1} - 8a_{n-2} \ \text{if} \ n > 1$$

**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.