Advanced High-School Mathematics

194 CHAPTER 4 Abstract Algebra

5. Show that if A, B, and C ⊆ U, and if A, B, and C are finite
   subsets, then

|A∪B∪C|=|A|+|B|+|C|−|A∩B|−|A∩C|−|B∩C|+|A∩B∩C|.

6. Try to generalize Exercise 5 above.^2


7. (Compare with Exercise 3 of Subsection 4.1.1) Consider the set
   S={ 1 , 2 , 3 , ..., 10 }, and define the sets

T = {ordered pairs (X,Y) of subsetsX,Y ⊆S, with|X|,|Y|=

2 andX∩Y =∅}

T′ = {subsets{X,Y}⊆ 2 S| |X|,|Y|= 2 andX∩Y =∅}

Compute|T|and|T′|.

8. In this problem the universal set is the real line R. Find A∪
   B, A∩B, A−B, B−A, and (A∪B)′, whereA= ]− 10 ,5] and
   B= [− 4 ,π].


9. In this problem the universal set is the Cartesian plane R^2 =
   {(x,y)|x, y∈R}. Define the subsets

A = {(x,y)|x^2 +y^2 < 1 } and B = {(x,y)|y≥x^2 }.

Sketch the following sets as subsets of R^2 : A∪B, A∩B, A−

B, B−A, and (A∪B)′.

10. LetA, B⊆Uand define thesymmetric differenceofAandB
    by setting

A+B = (A∪B)−(A∩B).

Using Venn diagram arguments, show the distributive laws

A+B = (A−B)∪(B−A)

(^2) This is the classical principle ofInclusion-Exclusion.