Information Technology Reference
In-Depth Information
Ta b l e 5 . 2
Examples of relations and operations (
iff
means if and only if)
Father-Son relation
F
(
x
,
y
)
iff
x
is father of
y
Son-Father-Grandfather relation
G
(
x
,
y
,
z
)
iff
x
is son of
y
who is son of
z
Arithmetical Order
the usual enumeration order
≤
over the natural
numbers
Number Divisibility
D
(
x
,
y
)
iff
x
can be exactly divided by
y
Linear betweeness
B
(
x
,
y
,
z
)
iff
point
y
is internal to the segment
x
,
z
Arithmetical operations
+
, − ,· ,/
Area measure
σ
(
P
)
is the area of a polygon
P
Number Factorization
factor
(
n
)
is the set of prime numbers dividing
n
Sequence occurrence
α
(
i
)
is the element at position
i
in sequence α
Sequence Length
|
α
|
is the length of α
and if
x
is the
image
of
x
, which belongs to
B
(
A
is called
domain
of
f
, while
B
is called
codomain
of
f
). The properties of
injectivity
,and
surjectivity
are expressed by the following formulae (where
∈
A
,then
f
(
x
)
⇒
is the logical implication):
x
=
y
∈
A
⇒
f
(
x
)
=
f
(
y
)
(
injectivity
)
y
∈
B
⇒
y
=
f
(
x
)
for some
x
∈
A
(
surjectivity
)
bijectivity
of
f
means both injectivity and surjectivity, that is, a correspondence one-
to-one between the set
A
and the set
B
. The imaging principle realized by functions
is one of the most powerful principles in mathematics, which can be found in a wide
variety of situations. In fact, a typical approach in mathematics is the representation
of certain objects (for example, physical states) by means of other objects, very
often more abstract (for example, numbers), in order to derive useful information
about the former ones by working with the latter ones. In some contexts it is useful
to consider
partial functions
, that is, functions that can be undefined on some el-
ements of their domains. However, unless explicitly stated, functions are implicitly
assumed to be total, that is, always defined on their domain.
Va r i a b l e s
are a crucial device of mathematical language. Any general statement
in mathematics uses variables implicitly or explicitly. For example, a geometrical
proof about triangles, begins with a preamble of such a type: “Given three points P,
Q, R, and a line passing through P ...”. In fact, in order to show any relationship
of general nature, it is essential to deal with “generic” elements. Generality means
variability, because a point P can be
instantiated
with any specific point when a
definite context is selected. Variables were essential for the development of algebra.
In fact, when we establish some conditions holding for a number, but we do not
know the exact identity of this number, we use a variable to denote it, and in this
sense that variable corresponds to a an
unknown
value. A
solution
of an equation in
one variable corresponds to the determination of a value of the variable that satisfies
the equality between the left hand side and the right hand side of the equation.
