204 CHAPTER 4 Abstract Algebra
Exercises
- LetS={ 1 , 2 , 3 , 4 }. How many relations are there onS?
- Let m∈Z+ and show that “≡ (modm)” is an equivalence rela-
tion onZ. How many distinct equivalence classes modmare there
inZ? - 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? - 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. - 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. - Suppose that we try to define a functionf :Z 4 → Zby setting
f([n]) =n−2. What’s wrong with this definition? - 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? - 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? - 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.