Advanced High-School Mathematics

196 CHAPTER 4 Abstract Algebra

see how the productS×S of two circles could be identified with the
torus(the surface of a doughnut)?^3

Finally, it should be obvious that ifAandBare finite sets|A×B|=
|A|·|B|.

Exercises

1. Let nbe a positive integer and let S={ 1 , 2 , ...,n}. Define the
   subsetT ⊆S×SbyT ={(a,b)∈S×S| |a−b|= 1}. Compute
   |T|as a function ofn.


2. Letnbe a positive integer and letSbe as above. Define the subset
   Z⊆S×S×SbyZ={(a,b,c)∈S×S×S|a, b, care all distinct}.
   Compute|Z|as a function ofn.


3. LetXandY be sets, and letC, D⊆Y. Prove thatX×(C∪D) =
   (X×C)∪(X×D).


4. LetXandY be sets, letA, B⊆Xand letC, D⊆Y. Is it always
   true that

(A∪B)×(C∪D) = (A×C)∪(B×D)?

5. LetT andT′be the sets defined in Exercise 7 of Subsection 4.1.2.
   Which of the following statements are true:

T ∈S×S, T⊆S×S, T ∈ 2 S, T⊆ 2 S

T′∈S×S, T′⊆S×S, T′∈ 2 S, T′⊆ 2 S

(^3) Here’s a parametrization of the torus which you might find interesting. LetRandrbe positive
real numbers withr < R. The following parametric equations describe a torus of outer radiusr+R
and inner radiusR−r:
x= (R+rcosφ) cosθ
y= (r+rcosφ) sinθ
z=rsinφ.