Expert: Scott A Wilson - 3/5/2010

Question
QUESTION: If an element or two from the set A are not paired with any element of set B, then is the domain of the relation the whole set A? or the set of only those elements that are paired?

ANSWER: I will assume that A has the domain and B has the range.

If not every element in A maps into B, then the domain is a proper subset of A.
The domain is the set of those elements from A that are paired.
The range is the set of those elements from B that are paired.


---------- FOLLOW-UP ----------

QUESTION: If A = {1, 2, 3}, B = {3, 4, 5, 6}, and the relation R = {(1,3), (2, 4)}then is it right to say that the domain of R = {1, 2} or A? Clearly, codomain is B and range = {3, 4}. I have a doubt only about the domain of R.

When we use the mapping (arrow ) diagram too , we show the association of 1, 2 of A with 3, 4, of B.

Then we write the codomain = B, range = f(A) = {3, 4}.

But will the domain be written as A or only {1, 2}?

Please clarify.

Thanks in advance.

Seeta

Answer
The domain of the function is which variables the function acts upon.
In this case, the domain of R is 1 and 2, or {1, 2}, which is contained in A,
but not all of A.

Also, the range of R is {3, 4}, which is contained in B, but is not all of B.

The domain of that function is {1, 2}, and that is contained in A, but is not A.
The range is {3, 4}, and that is contained in B, but is not B.

Both of them are said to be proper subsets of the other.