Enumerate elements of the following relations from the set A Literally the first homework question, and I seem to be struggling. There doesn't seem to be any examples in our book, so I'm hoping someone might help walk me through it. I'm guessing it's pretty simple too...
I'm not looking for the answer, I'd just like some pointers on what I need to do, and how I should go about enumerating the elements.

  
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*Enumerate the elements of the following relations from the set A of positive integers less than or equal to 10 to the set B of positive
  integers less than or equal to 30.
  
  
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*An element a of A is related to the element b of B if b = 3 × a
R=(1,3)(2,6)(3,9)(4,12)(5,15)(6,18)7,21)(8,24)(9,27)(10,30)
  
*An element a of A is related to the element b of B if b = 2 × a - 1
R=(1,1)(2,3)(3,5)(4,7)(5,9)(6,11)(7,13)(8,15)(9,17)(10,19)

At first I thought I'd just have the two sets:
A = {1,2 3 ,4 5 ... 10 }
B = {1, 2, 3, 4, 5 ... 30 }
so would I just take a (lets say 1) and b (1 again)
1 = 3 * 1 (obviously false)? and continue on for each element in both sets? 
3 = 3 * 1 (true)
It just seems a little tedious to go through all combinations to test.
 A: The notation your book uses may be different, but it is common to denote a relation by something like "$aRb$" to represent that $a$ is related to $b$.
The first set of relations would look like $1R3$, $2R6$, etc.
Saying that two things are related is not the same as saying they're equal -- there are equivalence relations, which behave similarly to the standard "$=$", but there are plenty of other relations that do not; for instance, the relation "$<$" on $\mathbb{N}$. A relation is just a set of ordered pairs, so it does not even need to have a simple English definition.
Also, it is important to note that by default, relations are not symmetric; it is true that in your first example, we have $1R3$, but we do not have $3R1$. Nor are they necessarily reflexive: here, $1$ is not related to $1$.
These should be comments, but I lack the reputation required to do that.
A: A binary relation between sets $A$ and $B$ is a subset of the cartesian product $A \times B :=\{(a,b): a \in A, b \in B\}$.  In your case, the set $A \times B$ contains 300 elements.  The first relation is the subset $\{ (a, 3a): a \in A\} = \{ (1,3), (2,6), \ldots, (10,30) \}$ consisting of 10 elements.  The second relation consists of the 10 elements $(1, 1), (2, 3), (3, 5), \ldots, (10, 19)$. 
One can also define binary relations on a single set $A$ to be any subset of $A \times A$.  
