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Mathematics LibreTexts

1.1.E: Problems in Set Theory (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 1.1.E.1

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(AB)C[x(AB) and xC][(xA or xB), and xC][(xA,xC) or (xB,xC)].

Exercise 1.1.E.2

Prove that
(i) (A)=A;
(ii) AB iff BA.

Exercise 1.1.E.3

Prove that
AB=A(B)=(B)(A)=[(A)B].
Also, give three expressions for AB and AB, in terms of complements.

Exercise 1.1.E.4

Prove the second duality law (Theorem 2(ii)).

Exercise 1.1.E.5

Describe geometrically the following sets on the real line:
 (i) {x|x<0}; (ii) {x||x|<1}; (iii) {x||xa|<ε}; (iv) {x|a<xb}; (v) {x||x|<0}.

Exercise 1.1.E.6

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 ab, noting that {a,a}= {a}. Also note that {a}a.]

Exercise 1.1.E.7

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)|(x2)2+(y+5)29};
(viii) {(x,y)|x22xy+y2<0};
(ix) {(x,y)|x22xy+y2=0}.

Exercise 1.1.E.8

Prove that
(i) (AB)×C=(A×C)(B×C);
(ii) (AB)×(CD)=(A×C)(B×D);
(iii) (X×Y)(X×Y)=[(XX)×(YY)][(XX)×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. ]

Exercise 1.1.E.9

Prove the distributive laws
(i) AXi=(AXi);
(ii) AXi=(AXi);
(iii) (Xi)A=(XiA);
(iv) (Xi)A=(XiA);
(v) XiYj=i,j(XiYj);
(vi) XiYj=i,j(XiYj).

Exercise 1.1.E.10

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).


1.1.E: Problems in Set Theory (Exercises) is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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