Relations Definition and Example I'm really struggling with grasping the concept of relations. Take this for example:
$\begin{array}{l}{\text { Let } A=\{a, b, c\} \text { and } B=\{1,2,3,4\} . \text { Then } \mathcal{R}_{1}=\{(a, 2),(a, 3),(b, 1),(b, 3),(c, 4)\}} \\ {\text { and } \mathcal{R}_{2}=\{(1, b),(1, c),(2, b),(3, c),(3, c),(4, a),(4, c)\} \text { are relations between } A \text { and } B . \text { Define } \mathcal{R}}\end{array}$
From lectures I have the following definition for a relation 

A relation on a set $X$ is a statement about all ordered pairs of $X$. For a given pair $(x,y)$, the relation might be true or false

The example given was the operator $<$ , I can see how this fits the definition. But, the first example I have now idea how that can be seen as a relation. What do the indexes represent?
Another (simpler) example, which I decided to exhaustively list its ordered pairs, in order to fully understand the definition:
$S=\{1,2,3\} \text { and } a \sim b \iff a=1 \text { or } b=1 $
If we take all ordered pairs we have:
$S \times S = \{(1,1),(2,2),(3,3),(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)\}$
I take that what $a \sim b$ represents would be a statement about whether two elements belong to a subset $S'$ of $S \times S$ , such that $$S' = \{(a,1),(b,1) \}, \text{ i.e } S' = \{(1,1),(1,2),(1,3),(2,1),(3,1)\}$$  If they do, then we can claim that $a \sim b$ or "a relates to b"
Is this correct? If so, then to prove that this is an equivalence relation would this mean the following:


*

*Reflexive : $ \forall \ s \in S , (s,s) \in S'  = \{(a,1) \}$

*Symmetric : $ \forall \ s, r \in S (s,r) \in S'' = \{(b,1),(a,1)\}$

*Transitive :  no idea
 A: The connection between the definition of relation (a statement about all ordered pairs of $X$) and the list of ordered pairs is like naming a function by its values.  So the relation can be characterized by the set of all pairs which are related.  That's what $\mathcal{R}_1$ and $\mathcal{R}_2$ are.  The indices are just there to differentiate the symbols for the two distinct relations.  We're not really talking about a list of relations.
You did exactly the right thing with your relation $\sim$ and your list $S'$.  Although following the text's notation it would be more conventional to call is $\mathcal{R}$:
$$
    \mathcal{R} = \{(1,1),(1,2),(1,3),(1,3),(2,1),(3,1)\}
$$
When checking the properties, you don't reduce the original set of pairs.  You need to check whether certain pairs are in $\mathcal{R}$, and if certain pairs are in $\mathcal{R}$ whenever certain other pairs are in it.


*

*Is $\mathcal{R}$ reflexive?  That's equivalent to checking if all three are satisfied: $(1,1) \in\mathcal{R}$, $(2,2) \in \mathcal{R}$, $(3,3) \in \mathcal{R}$.

*Is $\mathcal{R}$ symmetric?  That's equivalent to seeing if the reversal of every pair in $\mathcal{R}$ is also in $\mathcal{R}$.

*Is $\mathcal{R}$ transitive?  That's sort of equivalent to seeing if, whenever two elements are related to a third element, they are related to each other (at least, it's equivalent if the relation is also symmetric).  I see that $(2,1) \in \mathcal{R}$ and $(1,3) \in \mathcal{R}$.  Is $(2,3) \in \mathcal{R}$?
