Advanced High-School Mathematics

(Tina Meador) #1

204 CHAPTER 4 Abstract Algebra


Exercises



  1. LetS={ 1 , 2 , 3 , 4 }. How many relations are there onS?

  2. Let m∈Z+ and show that “≡ (modm)” is an equivalence rela-
    tion onZ. How many distinct equivalence classes modmare there
    inZ?

  3. LetT 5 be the set of all 2-element subsets of{ 1 , 2 , 3 , 4 , 5 }and say
    define the relationRonT 5 by stipulating thatA 1 RA 2 ⇔A 1 ∩A 2 =
    ∅. Compute|R|. Which of the three properties of an equivalence
    relation doesRsatisfy?

  4. Let f :S→ T be a mapping and define a relation on Sby stip-
    ulating that sRs′ ⇔ f(s) = f(s′). (Note that this is says that
    sRs′ ⇔ sands′are in the same fibre of f.) Show that Ris an
    equivalence relation.

  5. Define the following relation on the Cartesian 3-spaceR^3 : PRQ⇔
    P andQare the same distance from the origin. Prove thatRis an
    equivalence relation on R^3 and determine the equivalence classes
    inR^3.

  6. Suppose that we try to define a functionf :Z 4 → Zby setting
    f([n]) =n−2. What’s wrong with this definition?

  7. Suppose that we try to define a function g :Z 4 → {± 1 ,±i} by
    settingg([n]) =in. Does this function suffer the same difficulty as
    that in Exercise 6?

  8. Suppose that we try to define function τ : Z → Z 4 by setting
    τ(n) = [n−2]. Does this function suffer the same difficulty as
    that in Exercise 6? What’s going on here?

  9. LetRbe the relation on the real line given byxRy⇔ x−y∈Z,
    and denote by theR/Rthe corresponding quotient set.^5 Suppose
    that we try to definep:R/R→Cby setting p([x]) = cos 2πx+
    isin 2πx. Does this definition make sense?


(^5) Most authors denote this quotient set byR/Z.

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