One-one functions

f : X → Y is one-one (injective) if different elements of X always have different images: for all x₁, x₂ ∈ X, f(x₁) = f(x₂) ⇒ x₁ = x₂. A function that is not one-one is called many-one.

To prove one-one, start from f(x₁) = f(x₂) and deduce x₁ = x₂. For f : N → N, f(x) = 2x: 2x₁ = 2x₂ gives x₁ = x₂, so f is one-one.

To disprove it, produce two different inputs with the same output. f : R → R, f(x) = x² is many-one because f(−1) = f(1) = 1.

The rule alone does not decide: x² is many-one on R and on Z (f(−2) = f(2)), but one-one on N, where there are no negative inputs. x³ is one-one on R.

Roll numbers of the students of a class give a one-one function from the class to N, since no two students share a roll number.

The greatest integer function, the modulus function and the signum function are many-one on R: [0.2] = [0.7] = 0, |−3| = |3| = 3 and sgn 2 = sgn 5 = 1.

Graph check for a real function: it is one-one exactly when no horizontal line meets its graph at more than one point.