Of course you don't see. You're an atheist. You're trying to debate Christians who have religious convictions and faith-based beliefs. They will tell you one thing, with their beliefs in mind, while you will tell them another, with -no- beliefs in mind. It's like comparing apples and oranges.. the two mindsets are completely different, almost to the point of being unable to be debated on the same page without severe difficulty.
I believe (heh

) that even if we follow certain rules laid out by religious teaching, all in all we are still beings of logic and reason, that can understand "earthly" things, even though we might not see the "big picture" behind it. So discussion is - as you say - at times difficult, but not impossible.
Your examples in this paragraph are flawed. You see, those relationships don't -just- have a lack of sex. They also have a lack of love, a lack of friendship, a lack of sharing, a lack of emotional bond of any kind, a lack of anything but convenience.
A house without a roof is not a real house to me. The object reffered to as "house" can also lack or not lack windows and a door, but that does not make my first statement invalid. If lack of one thing makes something wrong, lack of n more things does not change that fact.
If you remove sex, what you have left is emotions and thoughts.. which are individual to each and every person. Just because you can't base a relationship off of emotions, emotional bonds, and a more mentally-based way of doing things, does not mean -no one- can. It just means you can't, for whatever reason..
Of course i can. I just don't call it a
relationship
Your own personal belief is that without sex, there is nothing left holding two people together.. you can't be with them physically, so the relationship is nothing. That's a sad outlook on life, from my point of view, but you are welcome to have it and I wish you much luck in it.
Yes it's nothing. but it doesn't bother me, as i never had too much problems finding fulflling sex *and* love in my partners.
Regarding your "proof": Cute. Horribly wrong and filled with bad math that hurts my head, but cute

First off, proving an *opinion* wrong is impossible. You can debate an opinion, and in discussion possibly reach consensus - or not - but you can't *prove* or *disprove* it.
Next is that you're basing your proof on wrong assumptions. You're replacing implications with equality (which works both ways) - pure calculus is not the apropriate method for handling logic. If you want to work with my statements, use them in their proper format:
1) "Sex AND NOT love => NOT relationship"
2) "Sex AND love => relationship"
3) "love AND NOT sex => NOT relationship"
Now, even if we disregard both these things. I'm not even going to explain in what way limiting the range of your proof to the mentioned values does to only *allow* the results you want...
It's as simple as that: Assign a value as a minimum for achieving "relationship". Take any number. Assign half that number to both love and sex. So only by adding sex+love together you reach relationship.
This of course is as misleading as your own proof (albeit mathematically correct) and it only goes to show that by twisting numbers you can "proof" anything. You can also assign the number for love to the number of relationship and thus "prove" that love is relationship again. Such a proof is nothing but meaningless.
Btw, i agree that it might have been misleading to ask for opinions on this thread. I always consider any thread open for discussion, so i'm sorry if you thought otherwise. I'll be happy to hear more opinions now, if you want, and leave discussing those matters to other threads.