Single-Valued Functions. A
single-valued function or single-valued mapping is a
mapping of the elements x ∈ X into elements y ∈ Y.
This is expressed as f : X → Y or 
. x ∈
X there exists a unique element of y such that f(x)
= y. The set X is the domain of the function, Y is the codomain,
(not to be confused with the range, which we introduce shortly). To
denote the value of a function on a particular element we can use any of the
notations: f(x) = y, f : x → y or
simply x → y. f is the identity map on X if f(x)
= x for all x ∈ X.
. x ∈
X there exists a unique element of y such that f(x)
= y. The set X is the domain of the function, Y is the codomain,
(not to be confused with the range, which we introduce shortly). To
denote the value of a function on a particular element we can use any of the
notations: f(x) = y, f : x → y or
simply x → y. f is the identity map on X if f(x)
= x for all x ∈ X.
Let f : X → Y
. The range or image of f is
f (X) = {y|y
= f (x) for some x ∈ X}.
The range is a subset of the
codomain. For each Z ⊆ Y, the inverse image of Z is defined:
f −1(Z)
≡ {x ∈ X | f(x) = z for some z ∈ Z}.
Examples.
- Finite polynomials, f (x) = ex,
are examples of single valued functions which map real numbers to real numbers.
and the exponential function, - The greatest integer function, f (x) = ⌊x⌋, is a mapping from R to Z. ⌊x⌋ is defined as the greatest integer less than or equal to x. Likewise, the least integer function, f(x) = ⌈x⌉, is the least integer greater than or equal to x.
The -jectives. A function
is injective if for each x1 ≠ x2, f (x1)
≠
f (x2). In other words, distinct elements are mapped
to distinct elements. f is surjective if for each y in the
codomain, there is an x such that y = f (x). If a
function is both injective and surjective, then it is bijective. A
bijective function is also called a one-to-one mapping.
Examples.
- The exponential function f (x) = ex, considered as a mapping from R to R+, is bijective, (a one-to-one mapping).
- f (x) = x2 is a bijection from R+ to R+. f is not injective from R to R+. For each positive y in the range, there are two values of x such that y = x2.
- f (x) = sin x is not injective from R to [−1..1]. For each y ∈ [−1..1] there exists an infinite number of values of x such that y = sin x.
Injective Surjective Bijective
Figure 1.1:
Depictions of Injective, Surjective and Bijective Functions
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