Q:

Questions regarding Elementary Set Notation

I am struggling with summation notation! I am attempting to perform the statements that are in bold.
$$\ A = \text{{0, 1, 2, 3, 4}} $$
$$\ B = \text{{3, 5}} $$
$$\ C = \left\{(i, j) : i \in A, j \in B, i + j =3 \right\}$$
$$\ D = \left\{ \left\{i,j \right\} : i \in A, j \in B, i+j = 3 \right\} $$
List all the elements of C
I understand that:
(i, j) is a tuple  (or ordered set)
(i,j) "such that" ... i is an element of A, j is an element of B
What I don't understand:
Is this an ordered set, nested inside an unordered set?
would the output be in (i1, j1), (i2, j2) format?
not sure what to do with the i+j=3 portion of the statement?
List all the elements of D
I understand:
{i, j} is an unordered set.  
What I don't understand:
Does (i,j) versus {i,j} make any significant changes?
Would the output be similar to that of C?
Any help is greatly appreciated. Thanks!

A:

What I don't understand:
  Is this an ordered set, nested inside an unordered set?

Yes. Each element of $C$ looks like $(a, b)$.

would the output be in (i1, j1), (i2, j2) format?

What's an "output"? You're just asked to list some elements, so give this list in whatever your preferred way of giving a list is. Use bullet points or whatever. There's nothing formal and set-notational going on here.

Does this mean that C = {0,3}?

Almost. It means that $C = \{(0,3)\}$. (This is set notation!) Alternatively, here is a list of all the elements in $C$, using bullet points because that's how I like to list things:

$(0,3)$

and that's it.

Does (i,j) versus {i,j} make any significant changes?

In this case, no. Can you think of examples where it might? What would happen if $A = B = \{0,1,2,3,4\}$, for example?

