Chapter 1 Flashcards
What are the three most important kind of mathematical sentences
Universal Statement, Conditional Statement, Existential Statement
A Universal Statement says that…
… a certain property is true for all elements in a set.
A Conditional Statement says that…
… if something is true, then another thing has also to be true.
A Existential Statement says that…
… given a property that may or not to be true, there is at least one thing for which the property is true.
What is a set?
A set is a collection of objects.
What ∈ means?
It means “belongs to”.
What the Axiom of extension says?
It says that the set is determined by what its elements are. The repetion or order of the elements doesn’t matter.
What are continuous objects?
Mathematical objects that have no space between them. Ex: the real number.
Whate are Discrete objects?
Mathematical objects that have space between them. Ex: integers.
How can we describe the set of elements “1”, “2”, “3” and “4” using the Set-Roster Notation?
{1, 2, 3, 4}
What is the model for the Set-Builder Notation…
{x E s | P(x)}
What is a subset?
Is a set which all its elements belongs to another set.
What is a proper subset?
Proper subsets are subsets of sets that are not themselves.
What A⊆B means?
It means A is a subset of B.
What is an Ordered pair?
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.
What is a tuple?
It is a finite ordered list of elements.
What is a n- tuple?
A tuple with n-elements in it.
What is an Ordered n-tuple?
Is a generalization of the ordered pair to a set with a finite number of elements.
What is the Cartesian product?
Is the product of two or more sets that forms a new set of ordered pairs.
What A X B means?
The Cartesian product of A and B.
What is a string?
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.
What is a relation between mathematical objects?
Is anything that can be said that connects two or more differents objects.
What 0 S 1 means?
It means that 0 and 1 are connect by the relation S.
How a relation can be thougth in terms of Cartesian products?
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.
Given the relation B in Q X T, what is the domaind and the co-domain of B?
The domain of B is Q
The co-domain of B is T
What is a function?
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.
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?
F(x) is the unique element of the co-domian B which is related to x by F. Is is read “f of x”.
What a graph is consisted of?
Two finites sets:
a nonempty set V(G) of vertices.
a set E(G) of edges.
What are edges associated with?
Edges are associated with a set consisting ofd either one or two vertices.
What are called the vertices associated with an edge?
The endpoint of that edge.
What is called the relation between the edge and its endpoints?
edge-endpoint function.
What is a loop?
An edge with only one endpoint.
What are parallel edges?
Two distinct edges with the same of endpoints.
What is an adjacent vertex?
Two vertices that are connected by the same edge.
How can a vertice be adjacent to itself?
By being connect by the same edge twice.
What are adjacente edges?
Two edges incidents on the same endpoints.
What is an isolated vertex?
A vertex that has no edges incident on itself.
What is a direct graph consisted of?
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.
What it means to say that in a direct graph, the edge e1 is associated with the pair (v, w)?
It means that e1 is a direct edge from v to w.
How can the degree of a vertice be calculated?
deg(v) = the number of edges that are incident in v. If the edge is a loop, it is counted twice.
