Is the relation {(2,3),(3,3),(4,2),(5,1)} a function with domain and co-domain {1,2,3,4,5}?

Question

I got a question marked incorrect, however, searching around, I found that the general consensus was that I got the answer correct. I promise that I am not asking you to do my homework as it has already been graded, but I really need to know the truth if I am to learn anything.

The task was to decide whether a relation is a function.

The one I got wrong:

• Problem:

The domain and co domain are {1,2,3,4,5} and the relation R is given by the set of ordered pairs: {(2,3),(3,3),(4,2),(5,1)}.

• my answer was that is was a function - and it was marked as incorrect

• my instructor said that is was not a function because 1 does not have an associated value

• my response was that 5 had the associated value of 1 so it was still a function

• her response was that 5 was clearly in the co domain.

From what I can tell, no indication was given as to what values in the set were in the domain or in the co domain simply by this: {1,2,3,4,5}

Was I incorrect in my finding that 1 was not the domain but the co domain and that 5 was in the domain based on the information provided?

Answer 1

You specified a function as follows: $$f(2)=3, f(3)=3, f(4)=2, f(5)=1$$

This is not completely defined on the domain, as $1$ is not sent anywhere.

If instead you misunderstood the usual way functions are defined, you might argue that you instead defined the function $$f(3)=2, f(3)=3, f(2)=4, f(1)=5$$ This would of course be wrong, but also still not be a function, because it does not send $5$ anywhere.

Also, in this problem, both domain and codomain are $\{1,2,3,4,5\}$.

Answer 2

A function, by definition, must be well defined and defined everywhere. Here the problem is with the later. Your relation is not defined everywhere.

A relation $\sim$ is defined everywhere if every element in the domain set is related to at least one element in the codomain.

i.e. $\forall x\in D \ \exists y\in C \ s.t. \ (x,y)\in R$ where $R$ is the relation, $D$ is the domain, and $C$ is the codomain.