1.1.E: Problems in Set Theory (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Prove Theorem 1 (show that x is in the left-hand set iff it is in the right-hand set). For example, for (d),
x∈(A∪B)∩C⟺[x∈(A∪B) and x∈C]⟺[(x∈A or x∈B), and x∈C]⟺[(x∈A,x∈C) or (x∈B,x∈C)].
Prove that
(i) −(−A)=A;
(ii) A⊆B iff −B⊆−A.
Prove that
A−B=A∩(−B)=(−B)−(−A)=−[(−A)∪B].
Also, give three expressions for A∩B and A∪B, in terms of complements.
Prove the second duality law (Theorem 2(ii)).
Describe geometrically the following sets on the real line:
(i) {x|x<0}; (ii) {x||x|<1}; (iii) {x||x−a|<ε}; (iv) {x|a<x≤b}; (v) {x||x|<0}.
Let (a,b) denote the set
{{a},{a,b}}
(Kuratowski's definition of an ordered pair).
(i) Which of the following statements are true?
(a) a∈(a,b); (b) {a}∈(a,b); (c) (a,a)={a}; (d) b∈(a,b); (e) {b}∈(a,b); (f) {a,b}∈(a,b).
(ii) Prove that (a,b)=(u,v) if a=u and b=v.
[Hint: Consider separately the two cases a=b and a≠b, noting that {a,a}= {a}. Also note that {a}≠a.]
Describe geometrically the following sets in the xy-plane.
(i) {(x,y)|x<y};
(ii) {(x,y)|x2+y2<1};
(iii) {(x,y)|max(|x|,|y|)<1};
(iii) {(x,y)|y>x2};
(iv) {(x,y)|y>x2};
(vii) {(x,y)||x|+|y|<4};
(vii) {(x,y)|(x−2)2+(y+5)2≤9};
(viii) {(x,y)|x2−2xy+y2<0};
(ix) {(x,y)|x2−2xy+y2=0}.
Prove that
(i) (A∪B)×C=(A×C)∪(B×C);
(ii) (A∩B)×(C∩D)=(A×C)∩(B×D);
(iii) (X×Y)−(X′×Y′)=[(X∩X′)×(Y−Y′)]∪[(X−X′)×Y];
[Hint: In each case, show that an ordered pair (x,y) is in the left-hand set iff it is in the right-hand set, treating (x,y) as one element of the Cartesian product. ]
Prove the distributive laws
(i) A∩∪Xi=⋃(A∩Xi);
(ii) A∪∩Xi=⋂(A∪Xi);
(iii) (∩Xi)−A=∩(Xi−A);
(iv) (∪Xi)−A=∪(Xi−A);
(v) ∩Xi∪∩Yj=∩i,j(Xi∪Yj);
(vi) ∪Xi∩∪Yj=∪i,j(Xi∩Yj).
Prove that
(i) (∪Ai)×B=⋃(Ai×B);
(ii) (∩Ai)×B=∩(Ai×B);
(iii) (∩iAi)×(∩jBj)=⋂i,j(Ai×Bi);
(iv) (∪iAi)×(⋃jBj)=⋃i,j(Ai×Bj).