Lecture 3: Equivalence Relations


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1 Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts tht come up ll the time in mthemtics. Tody s concepts re the ides of sets nd equivlence reltions: 1 Sets A set, for the purposes of this lecture, is just some collection of ojects 1. We usully denote set y listing its elements in etween pir of curly rces {. For exmple, {1, 2, 3 is the set contining the numers 1, 2 nd 3, while {1, 2, slmon contins the numers 1 nd 2, long with slmon. We will often give these sets nmes, nd write things like A = {1, 2, slmon so tht we cn refer to the set contining 1, 2, nd slmon without hving to write out ll of the things in tht set every time. We cll the ojects tht mke up set the elements or memers of tht set. If we wnt to sy tht given oject is in set, we express this with the symol, pronounced in. For exmple, we write things like 2 A to express the notion tht 2 is n element of the set A we defined erlier. Sometimes, we will wnt to define set without writing down ll of the elements in the set. In these cses, we cn insted define set y writing down rule tht determines whether or not given numer is memer of tht set. For exmple, we cn t define the set of nturl numers N y writing down every element in N: there re infinitely mny elements we d hve to write! Insted, wht we cn do is give rule tht determines whether numer is in N: nmely, numer is in N if it is whole numer tht is nonnegtive. Formlly, we write this s N = { N exctly whenever n is nonnegtive whole numer. The rule tht we re proposing for our set is nturl numer precisely whenever is nonnegtive nd whole numer goes on the right of the verticl r. On the left of the r, we put the vrile, so tht when we re reding our rule we know wht letter corresponds to the elements of our set. Strictly speking, the prt on the left of this verticl r isn t necessry for understnding wht s going on in this nottion; the rule we ve written tells us everything we re looking for! However, it mkes our life esier to hve reminder efore we red our rule tht the vrile we cre out is. This is thing you ll run into lot in future mth/physics clsses: it s often s importnt to mke your nswers nd work esily understood s it is to mke it correct. Eventully, the ides we strt grppling with in the sciences re t the limits of humn comprehension; rekthrough in 1 If you go further off into mthemtics nd the field of set theory, it turns out tht this definition reks down in some firly strnge nd unexpected wys: you cn construct sets tht wind up doing remrkly wful things if you think of them s just ritrry collections! This isn t the point of this lecture, ut if you re interested I recommend checking out the wikipedi rticle on Russell s prdox for more informtion. 1
2 nottion tht simplifies the concepts t hnd cn sometimes e more vlule thn dozen new discoveries! It is possile to write set in mny different wys. For exmple, we could write N s the set N = { is either equl to 0, or there is some other numer N such tht = + 1. This definition is nice ecuse it doesn t rely on reder lredy knowing wht whole numers or nonnegtive numers re; insted, it simply defines nturl numer s something tht is either 0, or something you cn get y dding 1 to nother nturl numer. So 1 is nturl numer, ecuse you cn get 1 y dding 1 to 0. With this oservtion, we cn see tht 2 is nturl numer, ecuse you cn get 2 y dding 1 to 1, nd we know tht 1 is nturl numer. Then we cn see tht 3 is nturl numer, ecuse we cn get 3 y dding 1 to 2, which we just showed ws nturl numer... nd so on nd so forth. Some textooks will often just write some of the elements in set, insted of giving rule tht descries the elements in the set, s wy of descriing the set in sitution where the set is lredy wellunderstood. For exmple, mny textooks will write N = {0, 1, 2, 3, 4,... Z = {... 3, 2, 1, 0, 1, 2, 3,... to descrie the nturl numers nd integers, respectively. 2 Equivlence Reltions Definition. Tke ny set S. A reltion R on this set S is mp tht tkes in ordered pirs of elements of S, nd outputs either true or flse for ech ordered pir. You know mny exmples of reltions: Equlity (=), on ny set you wnt, is reltion; it sys tht x = x is true for ny x, nd tht x = y is flse whenever x nd y re not the sme ojects from our set. Mod n ( mod n) is reltion on the integers: we sy tht x y mod n is true whenever x y is multiple of n, nd sy tht it is flse otherwise. Less thn (<) is reltion on mny sets, for exmple the rel numers; we sy tht x < y is true whenever x is smller numer thn y (i.e. when y x is positive,) nd sy tht it is flse otherwise. Bets is reltion on the three symols (rock, pper, scissors) in the gme Rock PperScissors. It sys tht the three sttements Rock ets scissors, Scissors ets pper, nd Pper ets rock re ll true, nd tht ll of the other pirings of these symols re flse. 2
3 In this clss, we will study specific clss of prticulrly nice reltions, clled equivlence reltions: Definition. A reltion R on set S is clled n equivlence reltion if it stisfies the following three properties: Reflexivity: for ny x S, xrx. Symmetry: for ny x, y S, if xry, then yrx. Trnsitivity: for ny x, y, z S, if xry nd yrz, then xrz. It is not hrd to clssify our exmple reltions ove into which re nd re not equivlence reltions: Equlity (=) is n equivlence reltions on ny set you define it on it trivilly stisfies our three properties of reflexivity, symmetry nd trnsitivity. Mod n ( check: mod n) is n equivlence reltion on the integers. This is not hrd to Reflexivity: for ny x Z, x x = 0 is multiple of n; therefore x x mod n. Symmetry: for ny x, y S, if x y mod n, then x y is multiple of n; consequently y x is lso multiple of n, nd thus y x mod n. Trnsitivity: for ny x, y, z S, if x y mod n nd y z mod n, then x y, y z re ll multiples of n; therefore (x y) + (y z) = x y + y z = x z is lso multiple of n, nd thus x z mod n. Less thn (<) is not n equivlence reltion on the rel numers, s it reks reflexivity: x x, for ny x R. Bets is not n equivlence reltion on the three symols (rock, pper, scissors) in the gme RockPperScissors, s it reks symmetry: Pper ets rock is true, while Rock ets pper is flse. Equivlence reltions re remrkly useful ecuse they llow us to work with the concept of equivlence clsses: Definition. Tke ny set S with n equivlence reltion R. For ny element x S, we cn define the equivlence clss corresponding to x s the set {s S srx Agin, you hve worked with lots of equivlence clsses efore. For mod 3 rithmetic on the integers, for exmple, there re three possile equivlence clsses for n integer to elong to: {... 6, 3, 0, 3, 6... {... 5, 2, 1, 4, 7... {... 4, 1, 2, 5,
4 Every element corresponds to one of these three clsses. The concept of equivlence clsses is useful lrgely ecuse of the following oservtion: Oservtion. Tke ny set S with n equivlence reltion R. On one hnd, every element x is in some equivlence clss generted y tking ll of the elements equivlent to x, which is nonempty y reflexivity. On the other hnd, ny two equivlence clsses must either e completely disjoint or equl, y symmetry nd trnsitivity: if the sets {s S srx nd {s S s Ry hve one element t in common, then trx nd try implies, y symmetry nd trnsitivity, tht xry; therefore, y trnsitivity, ny element in one of these equivlence reltions must e in the other s well. Consequently, these equivlence clsses prtition the set S: i.e. if we tke the collection of ll distinct equivlence clsses, every element of S is in exctly one such set. 3 The Rtionl Numers One prticulrly useful use of the concept of equivlence clsses is the definition of the rtionl numers! In prticulr, sk yourself: wht is the set of the rtionl numers? Most people will quickly sy something equivlent to the following: {, Z, 0. The issue with this s set is tht it hs lots of different entries for numers tht we usully think re not different ojects! I.e. the set ove contins 1 2, 1 2, 2 4, 3 6, 4 8,..., ll of which we think re the sme numer! People usully then go ck nd chnge our definition ove to the following: {, Z, > 0, GCD(, ) = 1. This fixes our issue from erlier: we no longer hve duplicted numers running round. However, it hs other issues: suppose tht you wnted to define ddition on this set! Nively, you might hope tht the following definition would work: + c d d + c =. However, for mny frctions, the output of this opertion is not n element of our new set! = = 50 { /, Z, > 0, GCD(, ) = 1. These difficulties tht we re running into with the rtionl numers come from the fct tht, prcticlly speking, they ren t set in most contexts tht we work with them! Rther, they re set with n equivlence reltion: 4
5 1. The underlying set for the rtionl numers: {, Z, The equivlence reltion: we sy tht = c d tht k = lc nd k = ld. if there re pir of integers k, l such 3. A rtionl numer is ny equivlence clss of our set ove under the ove equivlence reltion. This is the ide we hve when we think of 1 2, 1 2, 2 4, 3 6, 4 8,... s ll representing the sme numer 1/2: we re identifying 1/2 with its equivlence clss! 4. In this setting, we define ddition, multipliction, nd ll of our other properties just how we would normlly: i.e. we define + c d d + c =, where the only wrinkle is tht y ech of, c d, d+c we ctully men tke ny element equivlent to these frctions, nd y equlity ove we ctully men our equivlence reltion. Actully proving this is n equivlence reltion is tsk we leve for the homework! Do it if you re interested. 5
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