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 Virgils 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 (18171862)
“Anyone who seeks for the true causes of miracles, and strives to understand natural phenomena as an intelligent being, and not to gaze at them like a fool, is set down and denounced as an impious heretic by those, whom the masses adore as the interpreters of nature and the gods. Such persons know that, with the removal of ignorance, the wonder which forms their only available means for proving and preserving their authority would vanish also.”
—Baruch (Benedict)
“In todays world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
—Urie Bronfenbrenner (b. 1917)