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

# Learning Target Quiz 3

:::info

This quiz contains *new versions of questions* for **Learning Targets 1 through 4** and *new questions* for **Learning Targets 5 through 8**.

* 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.

:::

## 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 $223$ to binary. *Show your work and circle your answer*.
2. Convert the base 8 integer $715$ to decimal. *Show your work and circle your answer*.
3. Convert the base 2 integer `11001100` to hexadecimal. *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 `11001010` and `10010111`. *Show your work and circle your answer*.
2. Subtract the base-2 integers `11001010` and `10010111`. *Show your work and circle your answer*.
3. Multiply the base-2 integers `11101` and `101`. *Show your work and circle your answer*.
4. Divide the base-2 integer `11001010` 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 $a$ divides $bc$, then $a$ divides $b$.*

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

**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.)

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

## :new: 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:

- $P(x)$: $x^3 > 0$
- $Q(x)$: $x$ is a multiple of 6

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) $P(-3)$
(b) $Q(1006)$
(c) $\exists x P(x)$
(d) $\forall y (\neg Q(x))$
(e) $\exists z Q(z)$
2. State the negation of the following without just inserting "Not" or "it is not the case that":
(a) $\forall x Q(x)$
(b) There is a maple tree in my backyard.
(c) All angles are acute.

**Success criteria:** All the answers in the first item are correct. At least two of the three partsof the second item are correct.

## :new: 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) $\{a \in \mathbb{N} \, : \, n^2 + 4 \geq 10 \}$
(b) $\{2^x \, : \, x \in \{1,3,5,7\}\}$

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) $\{0, \frac{1}{3}, \frac{2}{3}, 1 \}$
(b) $\{2, 6, 10, 14, 18, \dots \}$

3. Consider the sets:
$$\begin{array}
AA &= \{x,y,z,t\} \\
B &= \{s,t,u,v,w,x\} \\
C &= \{s,t,u,v\} \\
D &= \{t,u,s,v\}
\end{array}
$$
For each statement below, state whether the statement is True or whether it is False.
(a) $x \in D$
(b) $D \subseteq B$
(c) $\{t,v\} \subseteq C$
(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.

## :new: 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:
$$A = \{1,2\} \qquad B = \{n \in \mathbb{N} \, : \, n \ \text{is a multiple of 3}\} \qquad C = \{2,4,6,8\}$$
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. $|C \setminus A|$
2. $A \cup C$
3. $A \cap B$
4. $A \times \{u,v\}$
5. ${\cal{P}} (A)$ (the power set of $A$)
6. $\overline{B}$ (the complement of $B$)

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