181 lines
3.5 KiB
Markdown
181 lines
3.5 KiB
Markdown
**Test Yourself**
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Page 28
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1. A universal statement asserts that a certain property is _______ for _______.
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2. A conditional statement asserts that if one thing _______ then some other
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thing _______.
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3. Given a property that may or may not be true, an existential statement
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asserts that _______ for which the property is true.
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**Solutions**:
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1. true, for all elements of a set.
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2. is true, also has to be true.
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3. there is at least one thing
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---
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**Test Yourself**
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Page 37
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1. When the elements of a set are given using the set-roster notation, the order
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in which they are listed _______.
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**Solution**
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does not matter.
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2. The symbol $\mathbb{R}$ denotes _______.
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**Solution**
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The set of all real numbers.
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3. The symbol $\mathbb{Z}$ denotes _______.
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**Solution**
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The set of all integers.
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4. The symbol $\mathbb{Q}$ denotes _______.
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**Solution**
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The set of all rational numbers.
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5. The notation $\{x | P(x)\}$ is read _______.
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**Solution**
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The set of all $x$ such that $P(x)$ is true.
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6. For a set $A$ to be a subset of a set $B$ means that _______.
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**Solution**
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Every element in $A$ is an element in $B$.
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7. Given sets $A$ and $B$, the Cartesian product $A \times B$ is _______.
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**Solution**
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The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.
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8. Given sets $A$, $B$, and $C$, the Cartesian product $A \times B \times C$ is
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_______.
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The set of all ordered triples, $(a, b, c)$ where $a \in A$ and $b \in B$ and
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$c \in C$.
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**Solution**
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9. A string of length $n$ over a set $S$ is an ordered $n$-tuple of elements
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$S$, written without _______ or _______.
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**Solution**
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parentheses; commas
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---
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**Test Yourself**
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Page 45
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1. Given sets $A$ and $B$, a relation from $A$ to $B$ is _______.
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**Solution**
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a subset of the Cartesian product $A \times B$
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2. A function $F$ from $A$ to $B$ is a relation from $A$ to $B$ that satisfies
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the following two properties:
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a. for every element $x$ of $A$, there is _______.
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b. for all elements $x$ in $A$ and $yr and $z$ in $B$, if _______ then _______.
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**Solution**
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a. for every element $x$ of $A$, there is _______.
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an element $y$ of $B$ such that $(x, y) \in F$
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b. for all elements $x$ in $A$ and $y$ and $z$ in $B$, if _______ then _______.
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$(x, y) \in F$ and $(x, z) \in F$; $y = z$
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3. If $F$ is a function from $A$ to $B$ and $x$ is an element of $A$, then
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$F(x)$ is _______.
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**Solution**
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a unique element of $B$ that is related to $x$ by $F$.
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---
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**Test Yourself**
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Page 57
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1. A graph consists of two finite sets: _______ and _______, where each edge is
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associated with a set consisting of _______.
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**Solution**
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a finite; nonempty set of vertices; a finite set of edges.
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2. A loop in a graph is _______.
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**Solution**
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An edge that has a single endpoint.
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3. Two distinct edges in a graph are parallel if, and only if, _______.
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**Solution**
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They share the same set of endpoints.
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4. Two vertices are called adjacent if, and only if, _______.
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**Solution**
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They are connected by an edge.
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5. An edge is incident on _______.
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**Solution**
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Each of its endpoints.
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6. Two edges incident on the same endpoint are _______.
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**Solution**
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adjacent
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7. Two edges incident on the same endpoint are _______.
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**Solution**
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isolated
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8. In a directed graph, each edge is associated with _______.
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**Solution**
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an ordered pair of vertices called its endpoints
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9. The degree of a vertex in a graph is _______.
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**Solution**
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the number of edges that are incident on the vertex, with an edge that is a loop
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counted twice.
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