**Test Yourself**

Page 28

1. A universal statement asserts that a certain property is _______ for _______.

2. A conditional statement asserts that if one thing _______ then some other
   thing _______.

3. Given a property that may or may not be true, an existential statement
   asserts that _______ for which the property is true.

**Solutions**:

1. true, for all elements of a set.

2. is true, also has to be true.

3. there is at least one thing

---

**Test Yourself**

Page 37

1. When the elements of a set are given using the set-roster notation, the order
   in which they are listed _______.

**Solution**

does not matter.

2. The symbol $\mathbb{R}$ denotes _______.

**Solution**

The set of all real numbers.

3. The symbol $\mathbb{Z}$ denotes _______.

**Solution**

The set of all integers.

4. The symbol $\mathbb{Q}$ denotes _______.

**Solution**

The set of all rational numbers.

5. The notation $\{x | P(x)\}$ is read _______.

**Solution**

The set of all $x$ such that $P(x)$ is true.

6. For a set $A$ to be a subset of a set $B$ means that _______.

**Solution**

Every element in $A$ is an element in $B$.

7. Given sets $A$ and $B$, the Cartesian product $A \times B$ is _______.

**Solution**

The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

8. Given sets $A$, $B$, and $C$, the Cartesian product $A \times B \times C$ is
   _______.

The set of all ordered triples, $(a, b, c)$ where $a \in A$ and $b \in B$ and
$c \in C$.

**Solution**

9. A string of length $n$ over a set $S$ is an ordered $n$-tuple of elements
   $S$, written without _______ or _______.

**Solution**

parentheses; commas

---

**Test Yourself**

Page 45

1. Given sets $A$ and $B$, a relation from $A$ to $B$ is _______.

**Solution**

a subset of the Cartesian product $A \times B$

2. A function $F$ from $A$ to $B$ is a relation from $A$ to $B$ that satisfies
   the following two properties:

a. for every element $x$ of $A$, there is _______.

b. for all elements $x$ in $A$ and $yr and $z$ in $B$, if _______ then _______.

**Solution**

a. for every element $x$ of $A$, there is _______.

an element $y$ of $B$ such that $(x, y) \in F$

b. for all elements $x$ in $A$ and $y$ and $z$ in $B$, if _______ then _______.

$(x, y) \in F$ and $(x, z) \in F$; $y = z$

3. If $F$ is a function from $A$ to $B$ and $x$ is an element of $A$, then
   $F(x)$ is _______.

**Solution**

a unique element of $B$ that is related to $x$ by $F$.

---

**Test Yourself**

Page 57

1. A graph consists of two finite sets: _______ and _______, where each edge is
   associated with a set consisting of _______.

**Solution**

a finite; nonempty set of vertices; a finite set of edges.

2. A loop in a graph is _______.

**Solution**

An edge that has a single endpoint.

3. Two distinct edges in a graph are parallel if, and only if, _______.

**Solution**

They share the same set of endpoints.

4. Two vertices are called adjacent if, and only if, _______.

**Solution**

They are connected by an edge.

5. An edge is incident on _______.

**Solution**

Each of its endpoints.

6. Two edges incident on the same endpoint are _______.

**Solution**

adjacent

7. Two edges incident on the same endpoint are _______.

**Solution**

isolated

8. In a directed graph, each edge is associated with _______.

**Solution**

an ordered pair of vertices called its endpoints

9. The degree of a vertex in a graph is _______.

**Solution**

the number of edges that are incident on the vertex, with an edge that is a loop
counted twice.