your opinion that to say we must have both (a,b) in R AND (b,a) in R in order to conclude something about the "relationship" between a and b is simply not true.
for example, let S = N, the natural numbers, with R = <: that is: aRb if and only if a < b.
well, 3 < 4, so (3,4) is an element of R. however, (4,3) is NOT in R, because 4 is NOT less than 3.
let's look more closely at statements of the form:
If A implies B, then C.
here is one statement of such a form:
"If rain falling on my car makes it wet, I shall wash it".
here: A = rain falls on my car, B = my car gets wet, C = I shall wash my car.
now given A, and given A implies B, we may conclude C.
but: this is not the ONLY circumstance under which C occurs: one might decide to wash their car just for the "heck of it", even if it has not rained.
also, it may be that "A implies B" is FALSE (perhaps the car is parked inside a garage when it rains), and yet the car is still washed at some point.
so we do NOT require A for C, or even A implies B, these are sufficient but not NECESSARY.
people often struggle with implication:
a statement like "P implies Q" in many people's minds means: we need to know P is true in order to know Q is true (we often use it this way in casual conversation).
but this is NOT what it means in mathematics: it means IF P is true, THEN Q is true (and if P is not true, we can't tell just from this information if Q is or isn't true).
a simple example:
if an integer is divisible by 4, it is an even number (divisible by 2).
this is a true statement.
but 6 is not divisible by 4, and yet 6 is nevertheless even, and
3 is not divisible by 4, and is NOT even.
so just because a number is not divisible by 4 (the premise is false), we cannot conclude that the consequence is true or false (it can go either way).
of course, statements like: "if the sky is green, then i'm a monkey's uncle" while "logically" true, are rather nonsensical: mathematical linguistics and natural language linguistics are not a perfect match. in mathematics "if...then", "and" and "or" have "special" meanings which are not quite the same as our everyday usage of them.