Courtesy of Mehran Sahami, Computer Science Department, Stanford University. Relations. G = { Zeus, Apollo, Kronos, Poseidon }

Transcription

1 Key topis: Reltions * Introdution nd Definitions * Grphs nd Reltions * Properties of Reltions * Equivlene Reltions * Prtil Orderings * Composition of Reltions * Mtrix Representtion * Closures * Topologil Sorting * PERT nd CPM; Sheduling Suppose we hve set of Greek deities: G = { Zeus, Apollo, Kronos, Poseidon } As everyone knows, Zeus is the fther of Apollo, Kronos is the fther of Poseidon, nd Kronos is lso the fther of Zeus. It seems tht there exists some omintion of the elements of G tht stisfy the "is the fther of" reltion. To express this more preisely: A reltion R etween two sets A nd B is suset of A X B. In other words, A X B produes set of ordered pirs <, >, oming from set A, nd from set B. Some of these pirs will e "interesting", i.e., will stisfy our reltion. For these pirs, we n write R, where R is the symol for our reltion. Alterntively, we n write <, > R. (Note tht R is generi reltion symol. Some reltions, suh s "less-thn" hve their own symol: <.) To e preise, wht we hve defined is inry reltion, so lled euse it opertes on ordered pirs. We n lso define unry reltions, whih operte on single elements, or ternry reltions, whih operte on ordered triples. In generl n n-ry reltion will operte on n-tuples. Exmple Let's onsider the is the fther of reltion (whih we will denote y F ) on the set G X G. We n figure out tht G X G = { <Zeus, Zeus>, <Zeus, Apollo>, <Zeus, Kronos>, <Zeus, Poseidon>, <Apollo, Zeus>, <Apollo, Apollo>, <Apollo, Kronos>, <Apollo, Poseidon>, <Kronos, Zeus>, <Kronos, Apollo>, <Kronos, Kronos>, <Kronos, Poseidon>, <Poseidon, Zeus>, <Poseidon, Apollo>, <Poseidon, Kronos>, <Poseidon, Poseidon> }

2 But of the set G X G, only suset stisfies the "is the fther of" reltion. Thus, pplying the F reltion to G X G, yields the set: { <Zeus, Apollo>, <Kronos, Poseidon>, <Kronos, Zeus> } mening tht Zeus F Apollo, Kronos F Poseidon, nd Kronos F Zeus. Grphs nd Reltions Grphs re generl representtion for expressing mny-to-mny reltionships. The esiest wy to understnd the definition of grph is to look t piture. The following re ll exmples of grphs: Intuitively, we get the ide tht grph is unh of points onneted y lines. The forml definition onveys this onept it more otusely: A grph is n ordered triple <N, A, f > where N is nonempty set of nodes or verties (dots) A is set of rs or edges (lines) f is funtion ssoiting eh r with n unordered pir x, y of nodes lled the endpoints of. Grphs re inredily useful strutures, nd the first use we will put them to is to represent the fmily reltion desried y the fther of reltion. Cronus Zeus Poseidon Apollo The stute reder my note tht this grph hs rrows rther thn lines onneting the nodes. Tht is euse the grph ove is tully speil type of grph lled direted grph. A direted grph (digrph) is n ordered triple <N, A, f > where N is nonempty set of nodes A is set of rs f is funtion ssoiting eh r with n ordered pir <x, y> of nodes lled the endpoints of.

3 Direted grphs re very useful for representing inry reltions, where the reltion R is represented y drwing n rrow from to. Properties of Reltions A reltion is lled reflexive on set S if it stisfies x R x for ny x in S. Tht is, <x, x> R for ny x S. Equlity is reflexive on integers. For ny integer n, it is lwys true tht n = n. In pitures, these reltions re reflexive: In grph of reflexive reltion, every node will hve n r k to itself. A reltion is lled irreflexive on set S if x R x is not stisfied for every x in S. Tht is, <x, x> R for ll x S. The less thn reltion is irrelexive on the integers. No numer is less thn itself. A reltion is symmetri on set S if whenever x R y holds, y R x holds s well. Tht is, if x S nd y S nd <x, y> R then <y, x> R. The "is siling of" reltion is symmetri. If Apollo is siling of Artemis, then Artemis is siling of Apollo. It is esy to tell if reltion is symmetri y looking t its grph: it s symmetri if every line hs n rrowhed t either end. d A reltion is ntisymmetri on set S if x S, y S, if <x, y> R, nd x y, then<y, x> R,

4 The "is the fther of" reltion is ntisymmetri. If Zeus is the fther of Apollo, then ertinly Apollo is not the fther of Zeus (this doesn t hppen even in Greek mythology). In terms of direted grph, whenever there is n rrow going out from n element, reltion is ntisymmetri if there is not n rrow oming k to tht element from the destintion of the first rrow. A reltion is trnsitive on set S if whenever x, y, nd z re in S, nd x R y holds, nd y R z holds, then x R z holds s well. Tht is, if x S, y S, z S, <x, y> R nd <y, z> R then <x, z> R. The "less-thn" reltion (<) is trnsitive. If x< y, nd y < z, then it must e true tht x < z. Exmple Consider the following reltions on {,,, 4}: r : {<,>, <,>, <,>, <,>, <,4>, <4,>, <4,4> } r : {<,>, <,>, <,> } r : {<,>, <,>, <,4>, <,>, <,>, <,>, <4,>, <4,4> } r4 : {<,>, <,>, <,>, <4,>, <4,>, <4,> } r5 : {<,>, <,>, <,>, <,4>, <,>, <,>, <,4>, <,>, <,4>, <4,4> } r6 : {<,4> } Whih reltions re reflexive? Whih reltions re symmetri? Whih re ntisymmetri? Whih re trnsitive? Consider the following proof: If inry reltion R is symmetri nd trnsitive, it is lso reflexive. PROOF: Let x nd y e memers of the domin of R (whih is symmetri nd trnsitive). For ny two elements x nd y in R, if xry we know tht yrx, y symmetry. By trnsitivity, we know tht xry nd yrx imply xrx. Sine x is n ritrrily hosen memer of R's domin, we hve shown tht xrx for every element in the domin of R, nd thus R is reflexive. Wht do you think of this proof? Equivlene Reltions These properties of reltions n ome in pkge dels, nd these pkges re given speil nmes. A reltion tht is reflexive, symmetri, nd trnsitive on set S is lled n equivlene reltion on S.

5 Exmple Suppose R is the reltion on set of strings suh tht R if nd only if length() = length(), where length(x) mens length of string x. Sine length() = length(), we n sy R nd therefore, R is reflexive. Suppose R, so length() = length(). Then, length() = length() so R is lso symmetri. Finlly, suppose R nd R ; this mens length() = length() nd length() = length(). We see tht length() = length() so R is lso trnsitive nd is n equivlene reltion. The following reltions re defined on the set of ll people. Whih re equivlene reltions? ) {(,) nd hve met} ) {(,) nd were orn in the sme yer} ) {(,) nd shre ommon prent} Grphs of equivlene reltions re divided into islnds of interonneted nodes, eh seprte sugrph representing n equivlene reltion. These islnds re lled prtitions. More formlly, prtition of set S is olletion of disjoint nonempty susets of S tht hve S s their union. f d h e g i Consider the following proof: Let R e inry reltion on set A nd suppose R is symmetri nd trnsitive; If, for every x in A, there is y in A suh tht xry, then R is n equivlene reltion. PROOF: Suppose x is prtiulr ut ritrrily hosen element of A. We know there is y suh tht xry. By symmetry, we know yrx nd y trnsitivity, xrx. Therefore, this reltion is reflexive nd s given, is lso symmetri nd trnsitive. Therefore, R is n equivlene reltion. Wht do you think of this proof? Prtil Orderings A reltion R tht is reflexive, ntisymmetri, nd trnsitive on set S is sid to define prtil ordering on S. A set S together with prtil ordering R is lled prtilly ordered set or poset. Consider the reltion "is n nestor of" on set of Greek deities. We will define the reltion in suh wy tht eh person hs himself s n nestor s well s his prents, prents of prents, et. Given this definition, it is ler tht the "is n nestor of" reltion is reflexive nd trnsitive. It is

6 lso ler tht the "is n nestor of" reltion is ntisymmetri. Therefore, this reltion defines prtil ordering on the set of Greek deities. Cronus Zeus Poseidon Apollo Exmple 4 Show tht ">=" is prtil ordering on the set of integers. Sine >=, this reltion is reflexive. If >= nd >=, then = whih shows this reltion is ntisymmetri. If >= nd >=, then >= so this reltion is trnsitive. Thus, ">=" is prtil ordering on the set of integers. Define reltion R on the set of integers s follows: For ll integers m nd n, mrn <=> every prime ftor of m is prime ftor of n. Is this prtil order? The grphs of prtil orderings n e firly omplex. (Being oth reflexive nd trnsitive produes lot of rs.) One wy to simplify these grphs is to use speil kind of representtion known s Hsse digrm. The generl proedure for reting Hsse digrm is s follows: ) drw the direted grph for the reltion; ) remove ll the loops t eh node whih must e there for reflexivity; ) Remove ll edges tht must e there for trnsitivity; 4) Arrnge eh edge so tht its initil node is elow its terminl node s indited y the direted edges; 5) Remove ll rrows on the direted edges (sine ll edges point upwrd towrd their terminl node). The Hsse digrm for the nestor reltion is shown elow: Apollo Zeus Poseidon Cronus Another exmple of Hsse digrm for the prtil ordering {<,> <= } on {,,,4} is given elow long with its direted grph.

7 Beuse posets re t lest semi-ordered, it is possile to pik out some extreme elements. In Hsse digrms, these elements re esy to spot - they re the top nd ottom elements: An element of poset (S, R) is mximl if it is not less thn ny element of the poset. In the nestors exmple ove, Apollo nd Poseidon re mximl elements nd Apollo is the gretest element. An element of poset (S, R) is miniml if it is not greter thn ny element of the poset. Kronos is the only miniml element in the nestors exmple; it is lso the lest element. A totl ordering is speil se of prtil ordering where every element of the set is relted to every other element. The Hsse digrm for totl ordering is out s simple s possile: it s stright line. The exmple given ove for <= is totl ordering. Suppose R is prtil ordering on set A. Elements nd of A re sid to e omprle if nd only if, either R or R. Otherwise nd re nonomprle. R is totl ordering if ll the elements in the reltion re omprle. Exmple 5 The Hsse digrm for the prtil ordering { <,> mod = 0 } on the set of { positive divisors of 4 } is given elow. This is not totl ordering euse 6 nd 4 (for exmple) re not omprle. The seond digrm for the prtil ordering { <,> mod = 0 } on the set of { positive divisors of 8 } is totl ordering.

8 Consider the following reltion R defined on the set S = {0, }: For ll ordered pirs (,) nd (,d) in S x S, (,) R (,d) <= nd <= d. Is this prtil ordering? If so, wht would the Hsse digrm look like? Composition of Reltions Suppose R is reltion on A X B nd S is reltion on B X C. The ompostion of R nd S, denoted S o R, is reltion on A X C given y S o R = {<,> A, C, nd there is some B suh tht R nd S.} This definition is rther dense, ut the onept is very intuitive. The pitures elow will mke things more ler. In this exmple, we onsider the reltion R linking the set of students A with their dorms in B nd the reltion S linking dorms with Resident Assistnts in C. Lur Rihrd Jmes Twin Toyon Brt Boppy Beth Buly Thor Amy Role Pete Peppy Rhond Rhrh A B C This represents the reltion R on A X B nd the reltion S on B X C. The omposition S o R links students in A with their RA s in C: Lur Brt Boppy Rihrd Beth Buly Jmes Thor Pete Peppy Amy Rhond Rhrh A C

9 This represents the omposition S o R on A X C. The esy wy to onstrut omposition S o R is for eh pir of elements <,> ( A, C), simply hek if there is pth from to through ny element of B. This is ompletely equivlent to the definition given ove. Exmple 6 Find S o R where R = {<,>, <,4>, <,>, <,>, <,4>} nd S = {<,0>, <,0>, <,>, <,>, <4,>} Another wy to look t omposition of reltions is: find the ordered pirs where the seond element of the ordered pir in R mthes the first element of the ordered pir in S. For exmple, <,> in R nd <,> in S produe the ordered pir <,> in S o R. S o R = { <,0>, <,>, <,>, <,>, <,0>, <,>} Mtrix Representtion For humns, it is often onvenient to view reltion s grph. Computers, on the other hnd, re not very good t looking t pitures. The method of hoie for omputer is to represent reltion s mtrix. A reltion R from set A to set B will hve A rows nd B olumns. A in row i nd olumn j of the mtrix mens tht the reltion holds for the i th element of A nd the j th element of B; 0 mens the reltion does not hold. (The order of elements in the sets A nd B doesn t mtter. We re just giving them n ritrry ut onsistent numering so tht we know where to look in the mtrix.) Exmple 7 To represent the fther of reltion on the set of Greek deities s mtrix, we will ritrrily impose the following numering on the set: = Zeus = Apollo = Kronos 4 = Poseidon Thus the mtrix n e represented s:

10 Exmple 8 A = {,,} B = {,} R = reltion from A to B ontining <,> if A, B nd >. Wht is the mtrix representing R? The ordered pirs of the reltion R = {<,>, <,>, <,> } so the mtrix is: To find the omposition of two reltions, we n simply perform vrition on mtrix multiplition. Rther thn multiplying eh individul pir of elements, we will tke the logil nd, nd rther thn summing ll the pirs, we will tke the logil or. This is esier to see with digrm:?????? d?? e?? f?? =??? ( nd d) or ( nd e) or?? ( nd f)??? Computing row, olumn Eh position in the nswer mtrix is omputed using the sme tehnique. Why does this tehnique work? There is no mgi involved. The mtrix produt simply does mthemtilly wht you do with your eyes when you look t the grph of reltion. It is just heking if there is pth from to ( A, C) using ny intermedite point in B. The nds hek for pir of onnetions <,> nd <,>, nd we re or ing the results euse ny B is OK. Exmple 9 Let R = { <,>,<,>,<,>,<4,> } nd S = {<,>,<,>,<,> }. Find the omposition S o R. First we n drw the grph of the reltions R nd S. (This is not neessry, ut it helps to show wht is going on.) Next to the reltions R nd S we hve drwn the omposition S o R.

11 4 4 The mtrix form of the reltions is shown elow = Closures If reltion R on set S fils to hve ertin property, we my e le to extend R to reltion R* on S tht does hve tht property. All this mens if we hve set of ordered pirs tht represent reltion R, we n dd ordered pirs to the reltion so the reltion eomes reflexive, symmetri, or trnsitive. If R* is the smllest suh set of ordered pirs, R* is lled the losure of R with respet to prtiulr property. Exmple 0 Let S = {0,,, } nd R = {<0,>, <0,>, <,>, <,>, <,>, <,0> }; R is not reflexive, symmetri or trnsitive. To mke this reltion reflexive, we must dd the ordered pirs <0,0> nd <,>. These pirs long with the originl pirs give us the losure of R with respet to reflexivity. Notie tht this is the smllest set of ordered pirs for reflexivity to hold. The losure of R with respet to symmetry is {<0,>, <0,>, <,>, <,>, <,>, <,0>, <,0>, <,0>, <,>, <0,> } Trnsitive losure is the sme s finding the omposition of R o R (ut there is little more to it thn tht s we shll see). We look for the ordered pirs where the seond element of the ordered pir mthes the first element of nother ordered pir. Thus, the trnsitive losure of R: R U {<0,0>, <0,>, <,0>, <,>, <,>, <,>, <,>} If A is the set of ll people nd R mens "is hild of", how would you desrie the trnsitive losure of R? Cn there e suh thing s ntisymmetri losure?

12 Trnsitive losures re espeilly useful in modeling ertin types of prolems. For exmple: Errti Airwys hs pulished the route mp shown elow. As n expert in suh mtters, you quikly relize tht the mp is tully direted grph, nd direted grph is equivlent to reltion. The reltion depited elow is onneted diretly y Errti Airwys on set of ities. STL DEN BOS SFO ORD IDC JFK ATL LAX A more useful reltion, however, might indite whether it ws possile to fly from one ity to nother y tking sequene of flights rther thn just one. In grph terms, the reltion onneted y sequene of flights is equivlent to determining whih pirs of nodes in the grph re onneted y pth. The reltion onneted y sequene of flights is relly the trnsitive losure of the originl reltion. This is more generl interprettion thn desried ove where we, in essene, were only looking for pths of length. The trnsitive losure is relly ll pths of ny length. Computing the trnsitive losure of ny resonly sized reltion y looking t the set of ordered pirs <x,y> n e nightmre, nd looking t the grph of the reltion does not help muh. For these tsks it is neessry to use some lgorithm. Fortuntely, we hve one hndy. (Note: the lgorithm in this hndout is not the est. We my see more effiient lgorithm lled Wrshll's lgorithm, when we study grph theory.) Consider the reltion R shown elow on the set S = {,,, 4}: 4 One wy to mke this trnsitive is to dd rs for ll the pths of length, nd then ll the pths of length, et until there re no more pths to e dded. But how n we find the pths of length? If we redrw R to reflet the ft tht it is reltion on S X S we see ll the pths of length.

13 4 4 To find ll the pths of length, we n redrw the reltion R next to itself We n now find ll the pths of length y inspetion of the grph ove. 4 4 As noted erlier, finding the trnsitive losure is the sme s finding the omposition R o R. Thus R o R gives ll pths of length, nd is sometimes denoted s R. This mens tht R U R ontins ll pths of length or less. To find ll pths of length, we ompute R = R o R o R = R o R = R U R U R Therefore, to find the trnsitive losure, we n use the following formul: R * = R U R U R... U Rn (n = # verties) Still, this is proly more work thn you wnt to do y hnd, so it s nie to hve the omputer do it for you using mtries. We lredy know how to ompose reltions using mtries. To find the "squre" of mtrix, you do mtrix multiplition. Then, the union of two mtries is simply the result of or ing the orresponding elements of eh mtrix. With this, it is possile to write progrm tht will find the trnsitive losure of ny reltion. Let R e the reltion on the set of ll students ontining the ordered pir (,) if nd re in t lest one ommon lss nd <>. When is (,) in R, R nd R *?

14 Topologil Sorting Prtil orderings hve n immedite pplition in mny sheduling lgorithms. It is esy to think of projets where some jos n e strted only fter some other jos hve een ompleted. (For exmple, the foundtion of uilding must e ompleted efore the 0 th story n e put on; efore the foundtion n e uilt, the site must e exvted, et.) These projets n e thought of s posets, where elements of set of jos re prtilly ordered y the reltion must e ompleted efore. The Hsse digrm in the first ox elow shows suh poset. In this exmple, f must e ompleted efore, whih must e ompleted efore, et. d d d e e e f g f d Extrting the totl ordering g,f,e,d,,, from poset. The trik to sheduling, of ourse, is to ome up with sequene of jos tht is onsistent with the requirements of the projet. In more forml terms, we re trying to find totl ordering tht is omptile with given prtil ordering. The topologil sort lgorithm is very strightforwrd solution. Given poset, we find miniml element. This is the first element in the totl ordering. We then remove this element from the poset nd repet finding miniml elements until there is nothing left. The oxes ove show the lgorithm in tion. Think of this in terms of rel projet, like king ke (or something like tht): g: put the owl on the ounter f: get ll the ingredients nd put them on the ounter e: throw ll the ingredients in the owl d: put the ingredients wy

15 : plug in the mixer : mix it ll up : throw it in the oven Notie tht there my e more thn one topologil sort for ny poset. We ould hve sorted: g, f,, e, d,,. This would hve worked just s well on the speified projet. Exmple The ove Hsse digrm represents the tsks needed to uild house. Do topologil sort to determine one possile order of tsks.

16 There re mny omptile totl orders. One possile nswer: Foundtion, Frming, Roof, Exterior Siding, Wiring, Pluming, Flooring, Wllord, Exterior Pinting, Interior Pinting, Crpeting, Interior Fixtures, Exterior Fixtures, Completion. PERT nd CPM Two widely used pplitions of prtil order reltions re PERT (Progrm Evlution nd Review Tehnique) nd CPM (Critil Pth Method). These tehniques me into eing in the 950s to del with the omplexities of sheduling the individul tivities needed to omplete very lrge projets. The tehniques re similr, ut they were developed independently. PERT ws developed y the US Nvy to help orgnize the onstrution of the Polris sumrine. CPM ws developed y DuPont Chemils for sheduling hemil plnt mintenne. The following exmple illustrtes how the tehniques work. Exmple At n uto ssemly plnt, the jo of putting together r n e roken down into the following tsks:. uild frme. instll, engine, power trin omponents, gs tnk. instll rkes, wheels, tires 4. instll dshord, floor, sets 5. instll eletril lines 6. instll gs lines 7. instll rke lines 8. tth ody to pnels 9. pint ody Some of these tsks n e done t the sme time s others, while some nnot e strted until others re finished. The following tle summrizes: Tsk Immeditely Preeding Tsks Time to omplete 7 hours 6 hours hours 4 6 hours 5, hours 6 4 hour 7, hour 8 4, 5 hours 9 6, 7, 8 5 hours We n uild Hsse digrm for this tle, ut the typil PERT or CPM representtion is to turn the digrm sidewys to reflet the hronologil orgniztion of tsks:

17 Wht is the minimum time required to uild r given unlimited resoures to do the work? You n determine this y working from left to right ross the digrm, noting for eh tsk, the minimum time required to omplete the tsk strting from the eginning of the ssemly proess. Tsk tkes 7 hours; Tsk requires the ompletion of tsk plus 6 hours for itself. Similrly, tsk tkes 0 hours. Tsk 5 requires the ompletion of tsks nd so the minimum time for tsk 5 = the time for tsk 5 itself + the mximum of the times to omplete tsks or = hours. So, tsk 5 tkes 6 hours. Similrly, tsk 7 tkes 4 hours, et. 6 hours is the minimum time to omplete the entire proess whih reflets the time to omplete sks,, 4, 8, 9. This pth is lled the ritil pth. Sheduling It is very rre tht given projet hs unlimited resoures to do the work. Given PERT hrt, ritil pth, nd some numer of resoures to do the work, how would you ome up with shedule over time showing the tsks the resoures re performing t eh step? We egin y showing time line with the ritil pth miniml time. For the ove exmple:

18 Then we drw in the tsks ssoited with the ritil pth, nd hve one person do these tsks. Finlly, we drw in the rest of the tsks ssuming tht one person n do ll of them. We will just strt numerilly with tsk. We fit it into the seond person's shedule wherever we n, sed on its dependenies. We do the sme with tsks 5, 6 nd 7. We end up with the following shedule: This exmple worked out esily, ut not ll do. You my find tht there just is no wy to omine the tsks so tht one person n do them ll. In this se, we dd nother resoure. Is there etter lgorithm thn this? There my e. If we find tht we n't fit tsk in, we n hek for other possile omintions. But if there re lrge numer of tsks, omputing n optiml omintion n tke very long time. It hs een proven tht optiml sheduling prolems re NP-Complete. So, our lgorithm (whih is nondeterministi) is the est we know of so fr. Severl sheduling softwre pkges use the PERT or CPM method. These pkges n e used to pln ny type of projet ut re espeilly gered towrd softwre development projets. You n input ll the tsks nd the dependenies etween tsks s well s sheduled times for eh tsk. Dependenies in rel projet inlude personnel issues s well s wht tsks must e ompleted to strt new tsk. It my e tht only one of your progrmmers n do prtiulr tsk. This n e refleted in the shedule nd estimted time to ompletion. So, you input the tsks (down to s detiled level s possile), the times for eh tsk, who is doing wht, wht it will ost for eh tsk, wht dys eh person hs off, et. The output is detiled pln ross time showing extly wht tsks re done when; who is doing them; how muh it will ost; tsk list for eh person involved, et. sed on the est shedule the softwre n ompute. As the projet progresses, the projet mnger keeps trk of where the "slippges" re, nd then retes new pln. At the end of the projet, lot of useful dt n e extrted from the estimted vs. tul omprisons. For exmple, perhps some memer of the tem onsistently used the prolems; or, perhps the originl estimtes were unrelisti; mye not enough time ws spent in design using lots more work in testing thn ntiipted. Biliogrphy Any good disrete mth ook hs setion on reltions. For more detiled informtion, refer to:

19 E.G. Coffmn, Computer nd Jo Sheduling Theory, New York: Wiley, 976. M. Grey, R. Grhm, D. Johnson, "Performne Gurntees for Sheduling Algorithms," Opertions Reserh, Vol. 6, 978. R. Grimldi, Disrete nd Comintoril Mthemtis, nd ed., Reding, MA: Addison-Wesley, 989. K. Rosen, Elementry Numer Theory nd its Applitions, nd ed., Reding, MA: Addison- Wesley, 988. S. Shni, Conepts in Disrete Mthemtis, Minnepolis, MN: Cmelot, 985. J.D. Wiest, F. Levy, A Mngement Guide to PERT/CPM, Englewood Cliffs, NJ: Prentie Hll, 98. Historil Notes The onept of reltions ws first introdued y Krl Friedrih Guss ( ) in Disquisitiones Arithmetie (whih he wrote t the ge of 4). In it, he disussed ongruene modulo m whih is defined s follows: If nd re integers nd m is positive integer, then is ongruent to modulo m if m divides -. This is notted _ (mod m). In other words, (mod m) if nd only if mod m = mod m. He lso proved (lthough using somewht different terminology) tht ongruene modulo m is n equivlene reltion. As mentioned ove, n equivlene reltion splits the elements of set into disjoint lsses lled prtitions. There re m different ongruene lsses modulo m, orresponding to the m different reminders possile when n integer is divided y m. These lsses form prtition of the set of integers. Consider for exmple, ongruene modulo 4: reminder 0: {..., -8, -4, 0, 4, 8,...} reminder : {..., -7, -,, 5, 9,...} reminder : {...,-6, -,, 6, 0,...} reminder : {..., -5, -,, 7,,..} These lsses re disjoint nd every integer is in extly one of them. This represents one of the first mppings of one set (the integers) to nother (ongruene modulo m) sed on reltion. Hsse Digrms were nmed fter Helmut Hsse ( ), Germn mthemtiin who introdued these digrms in his 96 textook Höhere Alger, s n id to solving polynomil equtions. Mtries were invented y Arthur Cyley (8-895) s mehnism for representing "higher spe" (spe of n dimensions).