your assumptions are "the commutative property is true" and "a * b = c".

No, dumbass. Those are not assumptions. The commutative property is true, a priori . This is not an assumption.

Likewise, "a * b = c" is not an assumption - it is a premise.

You seem to think that "premise" is synonymous with "assumption" which is why you look so fucking ridiculously stupid right now. Were you skullfucked so much as a child that your brain didn't develop correctly? Or what?

Were you angry?

HAHAHAHAHHAHAHAHAHAHAHAHA look at what the sad little Buttcoiner resorted to. The infamous "U MAD BRO?" tactic.

Whenever the "U MAD BRO" card gets pulled out in /r/Buttcoin, it is an undeniable signal that I have embarrassed the other person badly enough that even they realize that all hope is lost. In this case, I suspect you're dimly aware that you have been, to put it lightly, MAD REKT. You don't know how to cope with this, so you follow in the footsteps of your retarded forefathers, and resort to a sad and tired meme. It's pathetic, but then again, most of you are.

The definition is still blurry to me.

Hoooooooooooly shit this is funny. You're still having trouble with this... even after I broke it down to a level of simplicity that a child could handle.

I fail to see the substantial difference between the trade volume of a currency and how widespread its usage is

That's because you're a dumbass.

I can start an exchange today that allows 0 fee trading, and then send 100 satoshis back and forth between two accounts under my control a billion times per day. My exchange is going to show a ridiculously high trade volume but the size of the Bitcoin network (i.e. "how widespread its usage is") would be entirely unaffected.

Seriously, dude. If you can't see how fucking stupid you look with every successive sentence you write, that's almost impressive.