Injective Function - Proving That Functions Are One-to-one

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 : XY. 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.

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