386
7 Variational Principles
set of numbers X is called the supremum or least upper bound of X , it is written
as sup X . The greatest lower bound of a set of numbers X is called the infimum or
greatest lower bound of X , it is written as inf X . The supremum and infimum are
both unique.
Let A, B be two sets, if all the elements in A belong to B, then A is called the
subset of B, it is written as A ⊆ B or B ⊇ A. If all the elements in A belong to
B, and all the elements in B also belong to A, namely A and B contain the same
elements, then A and B are called equality, it is denoted by A = B. Obviously,
the sufficient and necessary conditions of A = B are A ⊆ B and B ⊆ A, namely
two equal sets are subset to each other. If A ⊆ B, but A = B, this shows that the
elements in A all belong to B, but in B least one element does not belong to A, then
A is called the proper subset of B, it is denoted by A ⊂ B or B ⊃ A. For any set
A, it is stipulated that Φ ⊂ A.
Let X be a set of functions, α, β are any real constants, if for any two functions
u, v in X , the function αu + βv also belongs to X , then X is called the linear set.
Two sets A, B by some calculations, can get another new set.
Let A, B be two nonempty sets, a set composed of all elements of A and all
elements of B is called the union set or union of sets of A and B, it is called the
union or join for short, it is denoted by A ∪ B, that is
A ∪ B = { x|x ∈ A or x|x ∈ B}
A set consists of the common elements that belong to both a set A and a set B is
called the intersection of sets of A and B, it is called intersection or meet for short,
it is denoted by A ∩ B, that is
A ∩ B = { x|x ∈ A and x|x ∈ B}
If the sets A and B have not the common elements, namely A ∩ B = Φ, then A
and B are called the mutually disjoint (sets), they are called the disjoint for short.
A set consists of the elements that belong to a set A but does not belong to a set
B is called the difference set of A and B, it is called the difference for short, it is
denoted by A − B or A\B, that is
A − B = A\B = { x|x ∈ A and x /
∈ B}
In studying a particular problem, If the considered set is the subset of a set X ,
then X is called the total set, universal set, basic set or fundamental set. If A is a
subset of X , then X − A is called the complementary set of A about X , is denoted
by A
C . A set consists of all subsets of a set X is called the power set of X .
Any two elements x and y form an ordered pair of elements, which is called the
ordered pair of x and y, it is denoted by (x, y). The elements x and y are called the
component of the ordered pair (x, y). The order refers when x = y, (x, y) = (y, x),
if (x 1 , y 1 ) = (x 2 , y 2 ), then x 1 = x 2 , y 1 = y 2 . The set of points defined by the ordered
pair (x, y) is called the phase space.
7 Variational Principles
set of numbers X is called the supremum or least upper bound of X , it is written
as sup X . The greatest lower bound of a set of numbers X is called the infimum or
greatest lower bound of X , it is written as inf X . The supremum and infimum are
both unique.
Let A, B be two sets, if all the elements in A belong to B, then A is called the
subset of B, it is written as A ⊆ B or B ⊇ A. If all the elements in A belong to
B, and all the elements in B also belong to A, namely A and B contain the same
elements, then A and B are called equality, it is denoted by A = B. Obviously,
the sufficient and necessary conditions of A = B are A ⊆ B and B ⊆ A, namely
two equal sets are subset to each other. If A ⊆ B, but A = B, this shows that the
elements in A all belong to B, but in B least one element does not belong to A, then
A is called the proper subset of B, it is denoted by A ⊂ B or B ⊃ A. For any set
A, it is stipulated that Φ ⊂ A.
Let X be a set of functions, α, β are any real constants, if for any two functions
u, v in X , the function αu + βv also belongs to X , then X is called the linear set.
Two sets A, B by some calculations, can get another new set.
Let A, B be two nonempty sets, a set composed of all elements of A and all
elements of B is called the union set or union of sets of A and B, it is called the
union or join for short, it is denoted by A ∪ B, that is
A ∪ B = { x|x ∈ A or x|x ∈ B}
A set consists of the common elements that belong to both a set A and a set B is
called the intersection of sets of A and B, it is called intersection or meet for short,
it is denoted by A ∩ B, that is
A ∩ B = { x|x ∈ A and x|x ∈ B}
If the sets A and B have not the common elements, namely A ∩ B = Φ, then A
and B are called the mutually disjoint (sets), they are called the disjoint for short.
A set consists of the elements that belong to a set A but does not belong to a set
B is called the difference set of A and B, it is called the difference for short, it is
denoted by A − B or A\B, that is
A − B = A\B = { x|x ∈ A and x /
∈ B}
In studying a particular problem, If the considered set is the subset of a set X ,
then X is called the total set, universal set, basic set or fundamental set. If A is a
subset of X , then X − A is called the complementary set of A about X , is denoted
by A
C . A set consists of all subsets of a set X is called the power set of X .
Any two elements x and y form an ordered pair of elements, which is called the
ordered pair of x and y, it is denoted by (x, y). The elements x and y are called the
component of the ordered pair (x, y). The order refers when x = y, (x, y) = (y, x),
if (x 1 , y 1 ) = (x 2 , y 2 ), then x 1 = x 2 , y 1 = y 2 . The set of points defined by the ordered
pair (x, y) is called the phase space.