1. Speaking Mathematically Flashcards
(24 cards)
A universal statement asserts that a certain property is ______ for ______.
A universal statement asserts that a certain property is true for all elements of a set.
- true
- all elements of a set
A conditional statement asserts that if one thing ______ then some other thing ______.
A conditional statement asserts that if one thing is true then some other thing also has to be true.
- true
- also has to be true
Given a property that may or may not be true, an existential statement asserts that ______ for which the property is true.
Given a property that may or may not be true, an existential statement asserts that there is at least one thing for which the property is true.
- there is at least one thing
When the elements of a set are given using the set-roster notation, the order in which they are listed ______.
does not matter
The symbol R denotes ______.
the set of all real numbers
The symbol Z denotes ______.
the set of all integers
The symbol Q denotes ______.
the set of all rational numbers
The notation {x | P(x)} is read ______.
the set of all x such that P(x)
For a set A to be a subset of a set B means that ______.
every element in A is an element is B
Given sets A and B, the Cartesian product A × B is ______.
the set of all ordered pairs (a, b) where a is in A and b is in B
Given sets A, B, and C, the Cartesian product A × B × C is ______.
the set of ordered triples of the form (a, b, c) where a ϵ A, b ϵ B, and c ϵ C
A string of length n over a set S is an ordered n-tuple of elements of S, written without ______ or ______.
parentheses ; commas
Given sets A and B, a relation from A to B is _____.
a subset of the Cartesian product A × B
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 y and z in B, if ______ then _______.
a. an element y of B such that (x, y) ϵ F (i.e., such that x is related to y by F).
b. (x, y) ϵ F and (x, z) ϵ F ;
y = z
If F is a function from A to B and x is an element of A, then F(x) is ______.
the unique element of B that is related to x by F
A graph consists of two finite sets: _____ and _____, where each edge is associated with a set consisting of _____.
a finite, non empty set of vertices
a finite set of edges
one or two vertices called its endpoints
A loop in a graph is _____.
an edge with a single endpoint
Two distinct edges in a graph are parallel if, and only if, ______.
they have the same set of endpoints
Two vertices are called adjacent if, and only if, ______.
they are connected by an edge
An edge is incident on ______.
each of its endpoints
Two edges incident on the same endpoint are _____.
adjacent
A vertex on which no edges are incident is _____.
isolated
In a direct graph, each edge is associated with ______.
an ordered pair of vertices called its endpoints
The degree of a vertex in a graph is ______.
the number of edges that are incident on the vertex, with an edge that is a loop counted twice.
