Originally Posted by

**oldguynewstudent** Q: Give an example of a function

f:{1,2,3,4} $\displaystyle \rightarrow$ {6,7,8,9}

which has no inverse.

A: x={1,2,3,4], y={6,7,8,9} f(x) = $\displaystyle \emptyset$

Marked wrong because f is not a funciton.

From Rosen:

DEFINITION 1: Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A.

Also from Rosen:

THEOREM 1: For every set S, (i) $\displaystyle \emptyset \subseteq$ S

and (ii) S $\displaystyle \subseteq$ S

If null is a subset of every set, isn't it an element in that set? And therefore f(x) = $\displaystyle \emptyset$ should be a function?

Thanks