In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c.
Transitivity is a key property of both partial order relations and equivalence relations.
Contents [hide]
1 Examples
2 Closure properties
3 Properties of transitivity
4 Other properties that require transitivity
5 Counting transitive relations
6 Examples
7 See also
8 Notes
9 References
10 External links
[edit]Examples

For example, "is greater than," "is at least as great as," and "is equal to" (equality) are transitive relations:
whenever A > B and B > C, then also A > C
whenever A = B and B = C, then also A = C
whenever A = B and B = C, then also A = C.
On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never be the mother of Claire.
Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation "is a matrilinear ancestor of". This is a transitive relation. More precisely, it is the transitive closure of the relation "is the mother of".
More examples of transitive relations:
"is a subset of" (set inclusion)
"divides" (divisibility)
"implies" (implication)
[edit]Closure properties

The converse of a transitive relation is always transitive: e.g. knowing that "is a subset of" is transitive and "is a superset of" is its converse, we can conclude that the latter is transitive as well.
The intersection of two transitive relations is always transitive: knowing that "was born before" and "has the same first name as" are transitive, we can conclude that "was born before and also has the same first name as" is also transitive.
The union of two transitive relations is not always transitive. For instance "was born before or has the same first name as" is not generally a transitive relation.
The complement of a transitive relation is not always transitive. For instance, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.
[edit]Properties of transitivity

For a transitive relation the following are equivalent:
Irreflexivity
Asymmetry
Being a strict partial order
[edit]Other properties that require transitivity

Preorder – a reflexive transitive relation
partial order – an antisymmetric preorder
Total preorder – a total preorder
Equivalence relation – a symmetric preorder
Strict weak ordering – a strict partial order in which incomparability is an equivalence relation
Total ordering – a total, antisymmetric transitive relation
[edit]Counting transitive relations

Unlike other relation properties, no general formula that counts the number of transitive relations on a finite set (sequence A006905 in OEIS) is known.[1] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in OEIS), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer[2] has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one is difficult. See also[3].
Number of n-element binary relations of different types
n all transitive reflexive preorder partial order total preorder total order equivalence relation
0 1 1 1 1 1 1 1 1
1 2 2 1 1 1 1 1 1
2 16 13 4 4 3 3 2 2
3 512 171 64 29 19 13 6 5
4 65536 3994 4096 355 219 75 24 15
OEIS A002416 A006905 A053763 A000798 A001035 A000670 A000142 A000110
[edit]Examples

equality
less/greater than (or equal)
inclusion of sets
inequality is not transitive
being an ancestor of is transitive
being a parent of is not transitive

1/3/2011 5:22:55 AM

I realize this thread is about transitive properties.... which I am not. Hear me out, maybe I can offer some insight.

I am in outside sales, which is currently salary+commission, but will move into straight commission starting at the beginning of July 2010. I have been in this position since July 2009. I have competition from several direct manufacturing sales reps, large distributors, and local distributors. Here are the advantages and disadvantages of each:

Direct Advantages: Immediate knowledge of new technology, no middle man mark up, one shipping bill (paid by manufacturer or buyer of goods), access to larger range of non-commodity items, control inventory, have access to many distributors that can effectively sell their goods which increases market share, and set prices of commodity they manufacture.

Direct disadvantages: Typically have 1-3 sales reps per region (i.e. southeast, mid-atlantic, northeast, etc.) limiting the number of accounts they can successfully manage/cold-call, lack physical customer service or physical technical service available to or affordable for smaller users or altogether, are sometimes not trustworthy because they will go in behind their distributors that sell their commodity to one account in large quantities (i.e. they missed a big account, and have found out about it through a distributor selling their particular product) which leads to the distributor not selling their product anymore, have too many distributors selling the product ultimately driving the set price down through deviations, possibly rely on distributors to actually sell the product, and competition from other direct sources.

Large distributor advantages: have access to other commodities that go hand in hand with other manufacturers (poor example- grocery stores sell milk as well as cereal), get direct pricing, many locations regionally or nationally easing the shipping burden of buyers with multiple locations, personal service either customer or technical, many sales reps that are able to cover a broader territory, access to multiple manufacturers of the same commodity allowing to keep prices in check, service programs that smaller companies can't offer and direct providers can't match in price or value, and experts of many many commodities as opposed to one or a few.

Large distributor disadvantages: smaller local distributors creating price wars (think Michael Scott Paper Co vs Dunder-Mifflin), direct mfg's going in behind and stealing business, limited access to all of the mfg's (you won't find Harris Teeter name brands in Food Lion and visa versa), can't truly set prices because it's based on both supply and demand, territory management, and tough growth prospects in slower economies (this is true for direct as well really)

Local distributor advantages: Typically a good ol' boy setting where the seller and the buyer know each other for years (this does happen at all levels, but mostly at the local level), local folks are right down the street and can be used in emergencies, if the local guy buys at high enough volumes then there is no shipping charge to the end user, and access to both direct mfg's and large distributors.

Local distributor disadvantages: easily beaten in price, array of commodities, array of technology, lack of trained staff, low cash flow, etc etc etc.

This is what I have noticed in my six months, I am sure there are plenty more that need mentioning. The way I am setting myself apart as a sales person is this: I go after the big accounts right now while I am new. The big accounts, if I land them, will take care of me while I am new and building a customer base. The money made off of those allows me to focus free time on smaller accounts that get me higher margins. I build up big accounts, I would like to have 5-10 of these, then get 20-30 medium accounts. If I lose 1 or 2 big accounts, the 20-30 medium accounts keep me afloat while I go after new big accounts. I don't really waste time on small accounts simply because they basically pay for breakfast or something really small.

I will say this, if you can't get a big account in the first 6-8 months (assuming you have cash flow that you can ride this long) you could be in a world of trouble. If you can get one, it will really make going after the others a lot more enjoyable and less stressful. It's simply just very exhausting wasting any time on anything other than big accounts in the very beginning. You work just as hard on the medium sized accounts and see 1/3 to 1/36 of the money in my situation.

If you have any other questions, you can PM me. I hope this helps in the slightest!

1/3/2011 5:42:14 AM

Lets dump everything we know about transitive property in here, TWW; technical, political, financial, everything is welcome. This in my new industry, I am investing all my money and time and I need your help. I've been taking classes in this for a while now and will be doing much more in the near future culminating in a new business venture to begin within a year. I have just broken my arm so I have lots of down time and can't really study that hard because of the meds, so lets just chat it up for the next few days. I especially hope to hear from all of the encouraging, helpful people from whom I've been receiving PM's. Lets get it all out there and discuss it, maybe even get some google docs going. By day, you can grind away at your engineering jobs but your heart can be here with me in Florida. Our mission is to reach critical mass and start a Transitive Revolution!

1/3/2011 6:58:06 AM

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1/3/2011 7:13:22 AM

^^which one was that? don't think i've seen it before.

1/3/2011 8:35:53 PM

pad

1/3/2011 8:42:21 PM

So I went out last night to White Collar Crime and ended up picking up a random chick on the dance floor that wanted to go back to my place with me. I happily oblige as she is a very attractive. I suggest a cab as this was my original plan to get home but she decides she wants to drive her car. We end up making it back to my apartment complex safe and park near one of the buildings. I inform her that she will be towed unless she parks in a visitor spot and in her inebriated state she decides to "stick it to the man" and parks in the grass between two buildings sevearal hundred feet away from the pavement. I promptly get her out of the car and I remove the car from the lawn and park it several buildings down as the only available visitor spots are there.

After that I walk back and hand the keys back to the girl before we go into my bedroom. We end up not doing anything so I grow bored quickly and decide to pass out. The girl tries to convince me several times to walk her out to her car but I refuse as I felt no obligation (It's not like I got anything). I tell her where her car is and fall asleep promptly as soon as she is gone.

Several hours later at 6 AM I am awakened by a Raleigh Police Officer shining a light in my eyes in my room (Roommate let him in after he was pounding on the door). He doesn't waste any time and starts interrogating me immediately. He claims I drove the vehicle into the grass (tire marks clearly present) and used a vehicle without "authorized use of transitive property." I of course deny both charges without telling the officer it was the girl as I wasn't sure what was going on and didn't want to get anyone in trouble. The officer then starts badgering me about where the keys to the vehicle were and the actual car. I quickly find the keys on the coffee table in the middle of the apartment and then lead him towards the extra visitor parking spots. Along the way he makes snide comments about taking a "nature walk" to have to find the vehicle, which is backed in neatly in the first visitor spot.

While walking outside I notice the girl is with the officer and can't figure out who the hell called the police. My best guess is a neighbor but as I think about it I am more and more I am sure it had to be the girl trying to find her car.

Anyway, after telling the officer multiple times I had nothing to do with the vehicle being parked on the grass, he decideds that it is indeed my fault (even if I didn't drive the vehicle) as I had invited a guest back to the apartment complex. He tells me he is going to charge me with a felony "unauthorized use of transitive property" unless I contact the apartment complex first thing this morning and let them know it was my fault as my guest had caused the damage. Evidently it is a crime to move a car off the grass and into a visitor parking spot.

In short, I am very pissed about this whole thing and am about to file a complaint against the officer for threatening me with charges that have no grounds. I'm not sure if it the best idea. Advice??

Note: I never had a chance to say anything to the girl, but after being awake for a few minutes I think she was actually the one who called the police in her drunken state, trying to find her vehicle. Hilariously she ends up in the back of the patrol car before the officer leaves. I still have no clue who called the police and led them to our apartment or if the girl ended up being charged with anything. I am assuming she recieved a DWI but I will never know as I can't remember her name and I told her I didn't want her number as she was leaving the first time as I felt she was a huge cock tease. Maybe I'll see it in the police blotter later. transitive property.

[Edited on January 3, 2011 at 8:52 PM. Reason : ]

1/3/2011 8:44:41 PM

^you're supposed to insert transitive property

1/3/2011 8:49:09 PM