Transformations are rules that convert one data into another data. We may denote a transformation by
T: X → Y,
where T is the transformation (also called map, function), X is the source set (or domain), and Y is the target set (or codomain). Taking any element x ∈ X as input, the transformation produces an element y = T(x) ∈ Y as output. We also call y the image (or value) of x under the transformation T. We may also include more specific information about the transformation by writing T: X → Y, x → y.
A more rigorous theory of transformations can be found here.
Example The following are some transformations in everyday life.
Age: people → number
IDs: student → number
IDp: professor → number
Instructor: course → professor
Maker: product → manufacturer
Capital City: country → city
Population: city → number
Example In mathematics, transformations are often given by formulae.
Square: R → R, x → x2
Reflection in x-axis: R2 → R2, (x, y) → (x, -y)
Polar-Cartesian: R2 → R2, (r, θ) → (rcosθ, rsinθ)
Root-Polynomial: R2 → R2, (x1, x2) → (x1 + x2, x1x2)
T: R3 → R2, (x1, x2, x3) → (x1 + 2x2 + 3x3, 4x1 + 5x2 + 6x3)
Here are two examples of mathematical transformations given by descriptions instead of formulae.
|Sign: R → R,||positive number → 1|
|negative number → -1|
|0 → 0|
|Dirichlet function: R → R,||rational number → 1|
|irrational number → 0|