Chapter 1

Question 1

What are the three most important kind of mathematical sentences

Answer 1

Universal Statement, Conditional Statement, Existential Statement

Question 2

A Universal Statement says that…

Answer 2

… a certain property is true for all elements in a set.

Question 3

A Conditional Statement says that…

Answer 3

… if something is true, then another thing has also to be true.

Question 4

A Existential Statement says that…

Answer 4

… given a property that may or not to be true, there is at least one thing for which the property is true.

Question 5

What is a set?

Answer 5

A set is a collection of objects.

Question 6

What ∈ means?

Answer 6

It means “belongs to”.

Question 7

What the Axiom of extension says?

Answer 7

It says that the set is determined by what its elements are. The repetion or order of the elements doesn’t matter.

Question 8

What are continuous objects?

Answer 8

Mathematical objects that have no space between them. Ex: the real number.

Question 9

Whate are Discrete objects?

Answer 9

Mathematical objects that have space between them. Ex: integers.

Question 10

How can we describe the set of elements “1”, “2”, “3” and “4” using the Set-Roster Notation?

Answer 10

{1, 2, 3, 4}

Question 11

What is the model for the Set-Builder Notation…

Answer 11

{x E s | P(x)}

Question 12

What is a subset?

Answer 12

Is a set which all its elements belongs to another set.

Question 13

What is a proper subset?

Answer 13

Proper subsets are subsets of sets that are not themselves.

Question 14

What A⊆B means?

Answer 14

It means A is a subset of B.

Question 15

What is an Ordered pair?

Answer 15

Given elements a and b (a, b) is an ordered pair meaning that a is the first element and d is the second element. To say (a, b) = (c, d), necessarily means that a = c and b = d.

Question 16

What is a tuple?

Answer 16

It is a finite ordered list of elements.

Question 17

What is a n- tuple?

Answer 17

A tuple with n-elements in it.

Question 18

What is an Ordered n-tuple?

Answer 18

Is a generalization of the ordered pair to a set with a finite number of elements.

Question 19

What is the Cartesian product?

Answer 19

Is the product of two or more sets that forms a new set of ordered pairs.

Question 20

What A X B means?

Answer 20

The Cartesian product of A and B.

Question 21

What is a string?

Answer 21

For a finite set A, a string of length n over A is an ordered n-tuple of elements of A written without the parentheses or commas.

Question 22

What is a relation between mathematical objects?

Answer 22

Is anything that can be said that connects two or more differents objects.

Question 23

What 0 S 1 means?

Answer 23

It means that 0 and 1 are connect by the relation S.

Question 24

How a relation can be thougth in terms of Cartesian products?

Answer 24

Let A and B be sets. A relation R from A to B is a subset of A X B. Given an ordered pair (x, y) in A X B, x is related to y by R, if, and only if (x, y) is in R. A is called the domain of R and B is called the co-domain.

Question 25

Given the relation B in Q X T, what is the domaind and the co-domain of B?

Answer 25

The domain of B is Q
The co-domain of B is T

Question 26

What is a function?

Answer 26

A function F from a set A to a set B is a relation that satisfies the following two properties:
1. For every element x in A, there is an element y in B such that (x,y) ∈ F.
2. For all elements x in A and y and z in B, if (x, y) ∈ F and (x, z) ∈ F, then y = z.

Question 27

If A and B are sets and F is a function from A to B, given an element x in A, what is F(x) and how is it read?

Answer 27

F(x) is the unique element of the co-domian B which is related to x by F. Is is read “f of x”.

Question 28

What a graph is consisted of?

Answer 28

Two finites sets:
a nonempty set V(G) of vertices.
a set E(G) of edges.

Question 29

What are edges associated with?

Answer 29

Edges are associated with a set consisting ofd either one or two vertices.

Question 30

What are called the vertices associated with an edge?

Answer 30

The endpoint of that edge.

Question 31

What is called the relation between the edge and its endpoints?

Answer 31

edge-endpoint function.

Question 32

What is a loop?

Answer 32

An edge with only one endpoint.

Question 33

What are parallel edges?

Answer 33

Two distinct edges with the same of endpoints.

Question 34

What is an adjacent vertex?

Answer 34

Two vertices that are connected by the same edge.

Question 35

How can a vertice be adjacent to itself?

Answer 35

By being connect by the same edge twice.

Question 36

What are adjacente edges?

Answer 36

Two edges incidents on the same endpoints.

Question 37

What is an isolated vertex?

Answer 37

A vertex that has no edges incident on itself.

Question 38

What is a direct graph consisted of?

Answer 38

Two finite sets:
A nonempty set V(G) of vertices
A set D(G) of directed edges, where each directed edge is associated with an ordered pair of vertices called its endpoints.

Question 39

What it means to say that in a direct graph, the edge e1 is associated with the pair (v, w)?

Answer 39

It means that e1 is a direct edge from v to w.

Question 40

How can the degree of a vertice be calculated?

Answer 40

deg(v) = the number of edges that are incident in v. If the edge is a loop, it is counted twice.