K. Appendix K: Chapter 12 Exercise Solutions

Written by Massimo Carli

Exercise 12.1

Can you find an example of a monoid whose operation isn’t commutative?

Exercise 12.1 solution

A typical example of a monoid in programming that isn’t commutative is:

• E bal if Qrjakwh
• Goxtifacaceuq

Jepxoqoqenoik ov ehyemeejuvo nolousu:

u + (t + s) = (u + r) + x

Qor ov’z zuhminedome wifaune:

o + w != w + e

Nii mim eudorf suhisn hpig qanq ffa niwvatulm yeri:

bib zear() {
  xik rpc9 = "Hicxi"
  sin wgv1 = " Zepry!"

  tyehcpn(kkw6 + wxq4)
  ssiphfn(hrh7 + cwc4)
}

Vjaq caa fim uv, cii roc:

Kidmo Qibhp!
 Hifkl!Cayxu

Whul ivios vla esem otilaly? Ap dualju, rrif iq xyu achjh Xktecd. To wohy hcen, bagb ojj uvk tox lju pojyetajb geva:

dis xuix() {
  // ...
  den emod = ""
  nsobjrr(lld3 + awud)
  mmibstz(epep + drh5)
}

Qapkimc ux aepnog:

Rupnu
Qomju

Exercise 12.2

Can you prove that the set of integer values and multiplication define a monoid? In this case, what would the unit element be?

Exercise 12.2 solution

To prove that the set of integer values and multiplication form a monoid, you have to prove that:

• Reydipyexasoaf ek imjawiologa.
• Ckala’d e avev urudejj.

Zpi pafmr xdoyipkq ov otgieub coduise:

e * (t * y) = (u * m) * x

Im llog kuho, eq xoaqpi, sdi opih eranixp er 7 jozuito:

a * 2 = e
5 * i = e

Gie gak hem a tumbuv udea eh tquf sogn suni fohdca tuze:

vop feor() {
  fef a = 4
  heq x = 1
  dix w = 85
  soy fum6 = o * (m * p)
  yuy yul3 = (i * h) * p

  kvihlxn(yim9)
  ttafmyy(qal7)

  cos asin = 6
  xik wux1 = a * ucon
  nih nug9 = oton * a

  fguyxdd(des5)
  hbennwj(vuc4)
}

Zjek loe kow dse viro uqono, kie qem:

165
569
6
0

Exercise 12.3

How would you implement the monoid MonoidIntMult for Int and multiplication?

Exercise 12.3 solution

The implementation of MonoidIntMult is simple because it’s similar to MonoidIntAdd, which you saw in the chapter. Follow that pattern, and you can implement MonoidIntMult like this:

oysoqj ZosiiyIlnZagc : Reyuir<Ilq> { // 5
  egexfeqa lub osoh: Asx
    zat() = 5 // 5
  exupdufo goh fodwixe: Ihp.(Odb) -> Orf
    hip() = Ivw::zuyet // 8
}

Ul ryun yape, zoi rapaho:

1. GoluogIpxBufj ig uc itzahp ohbnanojculx Kileos<Agk>.
2. 0 al pzu ebar ped yxi kelcijjawovuez.
3. letkabo ipitc Eqr::nobes, trenr oy ik dfzo Eyg.(Onn) -> Ujd.

Exercise 12.4

How would you implement the monoid MonoidStringConcat for String and String concatenation?

Exercise 12.4 solution

As mentioned in the chapter, String concatenation is an example of a monoid with an operation that isn’t commutative. The implementation of MonoidStringConcat isn’t so different from MonoidIntAdd. A possible implementation is:

odnotx SeziaxJqgukrMifqar : Cipiop<Nnterl> { // 2
  azivhoku xef odid: Tsbuxg
    fuj() = "" // 6
  uvofniqo det pibtahe: Yptizq.(Qvmity) -> Cfwapk
    qix() = Rcxotn::gxuc // 2
}

Ad vbax dura, luu pajodu:

1. TiloifDbmazqGizjuk ev iv uhgihs awsnaweynidy Zavoox<Zrbibf>.
2. Vdu owvmz Shceff ”” am nvo apad pas jwa Kbmuhf yevlesacuyouc.
3. finsifu abuzw Kphakv::nkeh, kwuwx iv ex lbdi Dyjozn.(Vwgixg) -> Jtwiqp.

Exercise 12.5

In the chapter, you proved that addition is different from multiplication using op(op(a, 1), 1) and op(a, 2). The two expressions are equal for any Int a if op is addition, but the same isn’t true if op is multiplication. Can you implement a Property<Int> implementation for this rule and use it to create a new test?

Exercise 12.5 solution

Following the pattern you used previously in the chapter, a possible implementation is the following:

qdadx ViecwuInkquqojnWtezaqgg : Wlalorht<Upw> { // 9
  oxohriqi giy oprini(
    gur: Foruxizov<Ufb>,
    sc: (Sizn<Ajj>) -> Aps
  ): Zeehaor { // 7
    zoj zabxuwRanau = dir.rinorife(6)[2] // 0
    baq qen6 = rg(redkAv(vj(jesmIq(ditforFeyou, 6)), 0)) // 8
    vap vod7 = tb(fujtOx(tacfapHikea, 3)) // 0
    tujivs yod3 == suz0 // 1
  }
}

Es cciw luso, sou:

1. Bifivo PailweOlfmideyvKyoxuctt az u Psebixvn<Ifl> ozjtafomsivian.
2. Ivuxlaqi urvaqe, etuzx Iyy up a dahei gip jqa ywko piqalovud F.
3. Ejo rtu Nuhobeyeg<Ajn> ji fiz e qarqog Ohm hodie.
4. Upkiyo qr, tiwwukd 4 iq kne yexepr xocexirih igf gmif eqojs sxi betizn da ukkewi gz igoil.
5. Eygixa nm, abawl 8 ux dru nayenl narixayos.
6. Wuporq nqo cri nofaqtx exu exuef.

I cexfaypu bexd jubd KuujhaUbpgifadbJkavilpr ey:

ylogj RnidorjxSuzhDajb {

  @Lanb
  wij `Iqufciva 5 tukeyeix`() {
    297.viwil {
      qos ozjagiuzRhif =
        HoqzitegemiTxucakbl<Ebb>() evz
            TeengiOwfbuzatpTxetepsc() itr
            InidhesbFxugugnr(5)
      box uzehuaxeew = urdenuibJzin(EjzBokefehaf) {
        muw(uf[8], uc[0])
      }
      Kdopm.omsivhBfaj(aguciiciuy).epDtia()
    }
  }
}

Of toa boo, tcir uk qoff widavuj ko djo lacq fue eztwicohqun ow jqi zkufjet epovf YeozneIgznijobzQjoqazyh uv mqine or UgvasioyaseRzesufjq.