Functions: one-one/many-one/into/onto

Functions can be classified according to their images and pre-images relationships. Consider the function x → f(x) = y with the domain A and co-domain B.

If for each x ε A there exist only one image y ε B and each y ε B has a unique pre-image x ε A (i.e. no two elements of A have the same image in B), then f is said to be one-one function. Otherwise f is many-to-one function.

e.g. x → x₃, x ε R is one-one function

while x → x₂, x ε R is many-to-one function. (see figure above)

e.g.  x = + 2, y = x₂ = 4

Graphically, if a line parallel to x axis cuts the graph of f(x) at more than one point then f(x) is many-to-one function and if a line parallel to y-axis cuts the graph at more than one place, then it is not a function.

        For a one-to-one function

        If x1 ≠ x₂ then f(x₁) ≠ f(x₂)

        or if (x₁) = f(x₂) => x₁ = x₂

One-to-one mapping is called injection (or injective).

Mapping (when a function is represented using Venn-diagrams then it is called mapping), defined between sets X and Y such that Y has at least one element 'y' which is not the f-image of X are called into mappings.

Let a function be defined as: f : X → Y

Where X = {2, 3, 5, 7} and Y = {3, 4, 6, 8, 9, 11}