Functions as Relations
Functions as Relations
Recall that a binary relation R from set A to set B is defined as a subset of
the Cartesian product A × B, which is the set of all possible ordered pairs (a, b),
where aϵ A and b ϵB .
If R ⊆ A × B is a binary relation and (a, b) ∈ R, we say that a is related to b by R.
It is denoted as aRb.
A function, denoted by f, is a special type of binary relation. A function from set A to
set B is a relation f ⊆ A × B that satisfies the following two properties:
Each element a ∈ A is mapped to some element b ∈ B.
Each element a ∈ A is mapped to exactly one element b ∈ B.
As a counterexample, consider a relation R that contains pairs (1, 1), (1, 2). The
relation R is not a function, because the element 1 is mapped to two elements, which
violates the second requirement.
In the next example, the second relation (on the right) is also not a function
since both conditions are not met. The input element 11 has no output value, and the
element 3 has two values - 6 and 7.
If f is a function from set A to set B, we write f : A → B . The fact that a
function f maps an element a ∈ A to an element b ∈ B is usually written
as f ( a )=b
Domain, Codomain, Range, Image, Preimage
, We will introduce some more important notions. Consider a function f :A→B.
The set A is called the domain of the function f , and the set B is the codomain. The
domain and codomain of f are denoted, respectively, dom( f ) and codom( f ).
If f ( a )=b , the element b is the image of a under f . Respectively, the element a is
the preimage of b under f . The element a is also often called
the argument or input of the function f , and the element b is called the value of the
function f or its output.
The set of all images of elements of A is briefly referred to as the image of A. It is also
known as the range of the function f , although this term may have different
meanings. The range of f is denoted rng( f ). It follows from the definition that the
range is a subset of the codomain.
Functions as Relations
Recall that a binary relation R from set A to set B is defined as a subset of
the Cartesian product A × B, which is the set of all possible ordered pairs (a, b),
where aϵ A and b ϵB .
If R ⊆ A × B is a binary relation and (a, b) ∈ R, we say that a is related to b by R.
It is denoted as aRb.
A function, denoted by f, is a special type of binary relation. A function from set A to
set B is a relation f ⊆ A × B that satisfies the following two properties:
Each element a ∈ A is mapped to some element b ∈ B.
Each element a ∈ A is mapped to exactly one element b ∈ B.
As a counterexample, consider a relation R that contains pairs (1, 1), (1, 2). The
relation R is not a function, because the element 1 is mapped to two elements, which
violates the second requirement.
In the next example, the second relation (on the right) is also not a function
since both conditions are not met. The input element 11 has no output value, and the
element 3 has two values - 6 and 7.
If f is a function from set A to set B, we write f : A → B . The fact that a
function f maps an element a ∈ A to an element b ∈ B is usually written
as f ( a )=b
Domain, Codomain, Range, Image, Preimage
, We will introduce some more important notions. Consider a function f :A→B.
The set A is called the domain of the function f , and the set B is the codomain. The
domain and codomain of f are denoted, respectively, dom( f ) and codom( f ).
If f ( a )=b , the element b is the image of a under f . Respectively, the element a is
the preimage of b under f . The element a is also often called
the argument or input of the function f , and the element b is called the value of the
function f or its output.
The set of all images of elements of A is briefly referred to as the image of A. It is also
known as the range of the function f , although this term may have different
meanings. The range of f is denoted rng( f ). It follows from the definition that the
range is a subset of the codomain.
