Proving That Functions Are One-to-one
A proof that a function ƒ is one-to-one depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the contrapositive of the definition of one-to-one, namely that if ƒ(x) = ƒ(y), then x = y. Here is an example:
- ƒ = 2x + 3
Proof: Let ƒ : X → Y. Suppose ƒ(x) = ƒ(y). So 2x + 3 = 2y + 3 => 2x = 2y => x = y. Therefore it follows from the definition that ƒ is one-to-one. Q.E.D.
There are multiple other methods of proving that a function is one-to-one. For example, in calculus if ƒ is differentiable, then it is sufficient to show that the derivative is always positive or always negative. In linear algebra, if ƒ is a linear transformation it is sufficient to show that the kernel of ƒ contains only the zero vector. If ƒ is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.
Read more about this topic: Injective Function
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—Marianne Moore (1887–1972)
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