discrete_mathematics_with_applications/chapter_1/test_yourself.md

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

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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

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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.

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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.