What does it mean to say that a function f A B is one to o

Solution

Suppose that f : A B is a function from A to B. If we pick a value y B, then x A is a pre-image of y if f(x) = y. Notice that I said a pre-image of y, not the pre-image of y, because y might have more than one preimage. For example, in the following function, 1 and 2 are pre-images of 1, 4 and 5 are pre-images of 4, 3 is a pre-image of 3, and 2 has no preimages.

link : http://mfleck.cs.illinois.edu/building-blocks/version-1.0/functions-one-to-one.pdf

u can check an image here if u want for counter example to one-one function

A function is one-to-one if it never assigns two input values to the same output value. Or, said another way, no output value has more than one preimage. So the above function isn’t one-to-one, because (for example) 4 has more than one pre-image. If we define g : Z Z such that g(x) = 2x. Then g is one-to-one. As with onto, whether a function is one-to-one frequently depends on its type signature. For example, the absolute value function |x| is not one-toone as a function from the reals to the reals. However, it is one-to-one as a function from the natural numbers to the natural numbers. One formal definition of one-to-one is: x, y A, x 6= y f(x) 6= f(y) For most proofs, it’s more convenient to use the contrapositive: x, y A, f(x) = f(y) x = y