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
Famous quotes containing the words proving that, proving and/or functions:
“Indeed, the best books have a use, like sticks and stones, which is above or beside their design, not anticipated in the preface, not concluded in the appendix. Even Virgil’s poetry serves a very different use to me today from what it did to his contemporaries. It has often an acquired and accidental value merely, proving that man is still man in the world.”
—Henry David Thoreau (1817–1862)
“One can prove or refute anything at all with words. Soon people will perfect language technology to such an extent that they’ll be proving with mathematical precision that twice two is seven.”
—Anton Pavlovich Chekhov (1860–1904)
“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)