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

# Learning Target Quiz 4

:::info

This quiz contains *new versions of questions* for **Learning Targets 1 through 8** and *new questions* for **Learning Targets 9 through 13**.

* 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.

:new: **Deadlines for your work:**

* Work on *Learning Targets 1-10* is due by **11:59pm ET Thursday, May 26**.
* Work on *Learning Targets 11-13* is due by **11:59pm ET Friday, May 27**.

:::

## 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 $197$ to binary. *Show your work and circle your answer*.
2. Convert the base 16 integer $7AF$ to decimal. *Show your work and circle your answer*.
3. Convert the base 2 integer `01101100` 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 `10111111` and `10100111`. *Show your work and circle your answer*.
2. Subtract the base-2 integers `10111111` and `10100111`. *Show your work and circle your answer*.
3. Multiply the base-2 integers `1001` and `101`. *Show your work and circle your answer*.
4. Divide the base-2 integer `10111111` by `101`. *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 $R$ is a field, then $R$ is a ring.* (Note, you do not need to know what those words mean.)

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

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

- $P(x)$: The ones digit of $x$ is even
- $Q(x)$: $x^2 = x$

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) $P(777)$
(b) $Q(2)$
(c) $\exists x Q(x)$
(d) $\forall y (\neg P(y))$
(e) $P(x)$
2. State the negation of the following without just inserting "Not" or "it is not the case that":
(a) $\exists x P(x)$
(b) All even numbers are prime numbers.

**Success criteria:** All the answers in the first item are correct. At least one of the two parts of the second item are 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) $\{x^3 \, : \, x \in \{0, 1, 2, 4\}\}$
(b) $\{ x \in \{0,1,2,4\} \, : \, x^3 \geq 10 \}$

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) $\{1, 10, 100, 1000, 10000, \dots \}$
(b) $\{2, 6, 10 \}$

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) $C \in A$
(b) $A \subseteq A$
(c) $a \in C$
(d) $\emptyset \subseteq A$
(e) $D = C$

:::info
*What "restate set membership in some way" means*: If your set is called $A$, you cannot "rewrite" $A$ as
$$A = \{ x \, : \, x \in A\}$$
For example if $A = \{0,1\}$, then the set-builder form of $A$ cannot be given as $\{ x \, : \, x \in \{0,1\}\}$. (It's correct, but trivial.) Instead restate set membership in some way, for example
$$A = \{ x \% 2 \, : \, x \in \mathbb{N} \}$$
This is not only correct but uses set-builder notation in a nontrivial way.
:::

**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.
:::

Consider the sets:
$$A = \{0,1,2\} \qquad B = \{n \in \mathbb{N} \, : \, n \, \% \, 4 = 1\} \qquad C = \{0, 1, 2, \dots, 10\}$$
The universal set $U$ is the set of all natural numbers.
Determine each of the following. If the answer is a set, write it in roster notation.

1. $|C|$
2. $A \cup C$
3. $C \cap B$
4. $A \times \{u,v\}$
5. $|{\cal{P}} (A)|$ (where ${\cal{P}} (A)$ the power set of $A$)
6. $C \setminus A$

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

## :new: 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. 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. $f: \{1,2,3\} \rightarrow \mathbb{N}$ given by $f(1) = 0$, $f(2) = 10$, $f(3) = 100$
2. The mapping $g: \{1,2,3,4\} \rightarrow \{x,y,z,t\}$ given by this matrix:
\begin{pmatrix}
1 & 2 & 3 \\
z & t & x
\end{pmatrix}
3. A computer language function `len` that accepts strings as inputs, and the output is the length of the string. For example `len('discrete structures') = 19` (the space is counted).

**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. $f: \{1,2,3,4,5\} \rightarrow \{0,1,2,3,4\}$ given by $f(a) = \lfloor a/2 \rfloor + 1$ where $\lfloor x \rfloor$ is the floor function.
2. $g$ is the function from the set of all 4-bit binary strings to the set $\{0,1,2,3,4\}$, defined by the following rule: Given a binary string for input, add up the bits (in base 10). For example $g(1011) = 3$.
3. $h: \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $h(n) = n + 1$

**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.
:::

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

2. Let $f$ be the function from the first part. Define $g:\{a,b,c,d\} \rightarrow \{1,2,3,4\}$ by $g(a) = 2, g(b) = 3, g(c) = 4, g(d) = 1$. Define $h: \{1,2,3,4\} \rightarrow \{a,b,c,d\}$ by $h(3) = a, h(4) = a, h(2) = c, h(1) = d$.
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) $(g \circ f)(1)$
(b) $(h \circ f)(3)$
(c) $(f \circ g)(x)$

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

## :new: 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. $\lfloor 1.1 \rfloor$
1. $\lfloor -1.1 \rfloor$
1. $\lceil 1.1 \rceil$
1. $\lceil -1.1 \rceil$
1. $6!$
1. $0!$
1. $DIV(50,20)$
1. $100 // 9$
1. $50 \ \% \ 7$
1. $6 \ \% \ 7$

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

## :new: 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 12-bit binary strings are there in all?
2. How many 12-bit binary strings are there, that have a `1` in the leftmost bit and `00` in the rightmost two bits? (One example of this is `101100100000`.)
3. How many 12-bit binary strings are there, that have *either* three `1` bits on the left end *or* two `1` bits on the right end?