# Thread: Definition of a function

1. ## Definition of a function

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

2. 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
It's a 'function' depending on how you define function. It is not a well-defined function which is what was assumed you were suppoesd to find. A function has to map element of $\displaystyle \text{Dom }f$ to exactly one element of $\displaystyle \text{Im }f$. Your 'function' satisfies the most commonly emphasized point that each element is mapped to at most one elment...but it fails to map each element to at least one element.

EDIT: A function $\displaystyle \phi:X\mapsto Y$ with $\displaystyle x\mapsto\varnothing$ is a function iff $\displaystyle X=\varnothing$ for the conditions of being a function are satisfied vaccuously since there is no $\displaystyle x$ in $\displaystyle \text{Dom }\phi$