7.1 Sets and Mappings
385
the sets are expressed in capital letters, the elements are expressed in lowercase
letters. The set taking set as element is called a class of sets, collection of sets or
family of sets. The set that contains no elements is called the empty set, null set or
vacant set, it is denoted by Φ. Any set other than the empty set is called nonempty
set or non-void set.
The set composed of points is called the point set or set of points. The set
composed of numbers called the set of numbers. The rest may be deduced by analogy.
Because the points on the number axis one to one correspond to all real numbers, the
number and point are not distinguished, the set of numbers and the point set are also
not distinguished. The set composed of real numbers is called the number line or
real line, it is usually denoted by R. The set composed of natural numbers is called
the set of natural numbers or natural numbers set, it is usually denoted by N.
The empty set or the set that only contains finite elements is called the finite set.
the set that contains infinite elements is called the infinite set. A set that has only
one element is called a singleton or one-element set. If the number of elements in
a finite set A is n, then n is called the count of A. It is stipulated that the count of
empty set Φ is zero.
There are two kinds of relationships between any element and a given set, namely
the element belongs to or does not belong to the set, which can only choose one
between the two. Let X be a set, when the element x belongs to X , it is denoted by
x ∈ X , when x does not belong to X , it is denoted by x /
∈ X .
The representation method of a set usually has two kinds, one is the enumeration
method, another is the characterization method. The enumeration method is suitable
for finite sets or infinite sets in which the elements can be arranged in a certain order,
the representation method is that the elements of a set are listed in the braces, for
example
X = {x 1 , x 2 , · · · , x n }; N = {1, 2, · · · , n, · · · }
The characterization method is that the properties or the conditions which should
be satisfied of the elements in a set are represented in the form of propositions, that
is
A set = {elements | properties or the conditions which should be satisfied of the
elements}
For example
R
n
= { x = (x 1 , x 2 , · · · , x n )| − ∞ < x i < ∞, i = 1, 2, · · · , n}
X = { x, y|(x − 3)
2
+ (y − 2)
2
= 16}
Let X be a set of numbers, M (or m) is a number, if for any x ∈ X , there is x M
or x m, then M is called an upper bound or a lower bound of X . Obviously, if X
has an upper bound or a lower bound, then it must have an infinite number of upper
or lower bounds, it is that all numbers greater than M can be the upper bound of X ,
all numbers less than m can be the lower bound of X . The least upper bound of a
385
the sets are expressed in capital letters, the elements are expressed in lowercase
letters. The set taking set as element is called a class of sets, collection of sets or
family of sets. The set that contains no elements is called the empty set, null set or
vacant set, it is denoted by Φ. Any set other than the empty set is called nonempty
set or non-void set.
The set composed of points is called the point set or set of points. The set
composed of numbers called the set of numbers. The rest may be deduced by analogy.
Because the points on the number axis one to one correspond to all real numbers, the
number and point are not distinguished, the set of numbers and the point set are also
not distinguished. The set composed of real numbers is called the number line or
real line, it is usually denoted by R. The set composed of natural numbers is called
the set of natural numbers or natural numbers set, it is usually denoted by N.
The empty set or the set that only contains finite elements is called the finite set.
the set that contains infinite elements is called the infinite set. A set that has only
one element is called a singleton or one-element set. If the number of elements in
a finite set A is n, then n is called the count of A. It is stipulated that the count of
empty set Φ is zero.
There are two kinds of relationships between any element and a given set, namely
the element belongs to or does not belong to the set, which can only choose one
between the two. Let X be a set, when the element x belongs to X , it is denoted by
x ∈ X , when x does not belong to X , it is denoted by x /
∈ X .
The representation method of a set usually has two kinds, one is the enumeration
method, another is the characterization method. The enumeration method is suitable
for finite sets or infinite sets in which the elements can be arranged in a certain order,
the representation method is that the elements of a set are listed in the braces, for
example
X = {x 1 , x 2 , · · · , x n }; N = {1, 2, · · · , n, · · · }
The characterization method is that the properties or the conditions which should
be satisfied of the elements in a set are represented in the form of propositions, that
is
A set = {elements | properties or the conditions which should be satisfied of the
elements}
For example
R
n
= { x = (x 1 , x 2 , · · · , x n )| − ∞ < x i < ∞, i = 1, 2, · · · , n}
X = { x, y|(x − 3)
2
+ (y − 2)
2
= 16}
Let X be a set of numbers, M (or m) is a number, if for any x ∈ X , there is x M
or x m, then M is called an upper bound or a lower bound of X . Obviously, if X
has an upper bound or a lower bound, then it must have an infinite number of upper
or lower bounds, it is that all numbers greater than M can be the upper bound of X ,
all numbers less than m can be the lower bound of X . The least upper bound of a