Solution Verification: Functions/Sets Question

Question

1. Which of the following is a true statement? (Assume a finite domain.)

a. If a function is not a one-to-one correspondence, its domain must contain more points than its image.

b. If a function is one-one, its domain and range are the same set.

c. If the domain of a function has more points than its image, it is not surjective.

d. If a function is onto, its domain and range are the same set.

B, C, D all seem false to me, but I am still unsure that the domain must contain more points than its image to satisfy not one-to-one correspondence.

B seems false because the sets need not be the same but just need the same number of points. Same with D.

C is false because even if the domain of a function has more points than its image, it can still be surjective.

This leaves me with A.

Answer 1

Apparently you’re to understand one-to-one correspondence as simply a one-to-one (injective) function, i.e., a one-to-one correspondence between its domain and its range, irrespective of whether the range is all of the codomain. On that understanding (a) is indeed true: the function can fail to be one-to-one only by sending two points of the domain to the same point of the range, and since the domain is finite, that means that the range must be smaller than the domain.

As you say, the other three are obviously false in general.

Answer 2

A is true: For every element in the domain, the function maps that element to no more than $1$ element in the range, i.e. maps it to $1$ element in the image. So it is impossible for the image to have more points than the domain. If the image has as many points as the domain, then it is one-to-one, i.e. injective. If the function is not injective, then the image has fewer points than the domain, and conversely, if the image has fewer points than the domain, the function is not injective (notice how both of these cases obviously violate the required one-to-one).

BCD are trivially false and their falseness can be verified by a quick look at the definitions for surjectivity (onto) and injectivity (one-to-one).