1.1.E: Problems in Set Theory (Exercises)
Prove Theorem 1 (show that \(x\) is in the left-hand set iff it is in the right-hand set). For example, for \((\mathrm{d}),\)
\[
\begin{aligned} x \in(A \cup B) \cap C & \Longleftrightarrow[x \in(A \cup B) \text { and } x \in C] \\ & \Longleftrightarrow[(x \in A \text { or } x \in B), \text { and } x \in C] \\ & \Longleftrightarrow[(x \in A, x \in C) \text { or }(x \in B, x \in C)]. \end{aligned}
\]
Prove that
(i) \(-(-A)=A\);
(ii) \(A \subseteq B\) iff \(-B \subseteq-A\).
Prove that
\[
A-B=A \cap(-B)=(-B)-(-A)=-[(-A) \cup B].
\]
Also, give three expressions for \(A \cap B\) and \(A \cup B,\) in terms of complements.
Prove the second duality law (Theorem 2(ii)).
Describe geometrically the following sets on the real line:
\[
\begin{array}{ll}{\text { (i) }\{x | x<0\} ;} & {\text { (ii) }\{x| | x |<1\}}; \\ {\text { (iii) }\{x| | x-a |<\varepsilon\} ;} & {\text { (iv) }\{x | a<x \leq b\}}; \\ {\text { (v) }\{x| | x |<0\}}. \end{array}
\]
Let \((a, b)\) denote the set
\[
\{\{a\},\{a, b\}\}
\]
(Kuratowski's definition of an ordered pair).
(i) Which of the following statements are true?
\[
\begin{array}{ll}{\text { (a) } a \in(a, b) ;} & {\text { (b) }\{a\} \in(a, b)}; \\ {\text { (c) }(a, a)=\{a\} ;} & {\text { (d) } b \in(a, b)}; \\ {\text { (e) }\{b\} \in(a, b) ;} & {\text { (f) }\{a, b\} \in(a, b)}. \end{array}
\]
(ii) Prove that \((a, b)=(u, v)\) if \(a=u\) and \(b=v\).
[Hint: Consider separately the two cases \(a=b\) and \(a \neq b,\) noting that \(\{a, a\}=\) \(\{a\} .\) Also note that \(\{a\} \neq a . ]\)
Describe geometrically the following sets in the \(x y\)-plane.
(i) \(\{(x, y) | x<y\}\);
(ii) \(\left\{(x, y) | x^{2}+y^{2}<1\right\}\);
(iii) \(\{(x, y)|\max (|x|,|y|)<1\}\);
(iii) \(\left\{(x, y) | y>x^{2}\right\}\);
(iv) \(\left\{(x, y) | y>x^{2}\right\}\);
(vii) \(\{(x, y)| | x|+| y |<4\}\);
(vii) \(\left\{(x, y) |(x-2)^{2}+(y+5)^{2} \leq 9\right\}\);
(viii) \(\left\{(x, y) | x^{2}-2 x y+y^{2}<0\right\}\);
(ix) \(\left\{(x, y) | x^{2}-2 x y+y^{2}=0\right\}\).
Prove that
(i) \((A \cup B) \times C=(A \times C) \cup(B \times C)\);
(ii) \((A \cap B) \times(C \cap D)=(A \times C) \cap(B \times D)\);
(iii) \((X \times Y)-\left(X^{\prime} \times Y^{\prime}\right)=\left[\left(X \cap X^{\prime}\right) \times\left(Y-Y^{\prime}\right)\right] \cup\left[\left(X-X^{\prime}\right) \times Y\right]\);
[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) \(A \cap \cup X_{i}=\bigcup\left(A \cap X_{i}\right)\);
(ii) \(A \cup \cap X_{i}=\bigcap\left(A \cup X_{i}\right)\);
(iii) \(\left(\cap X_{i}\right)-A=\cap\left(X_{i}-A\right)\);
(iv) \((\cup X _{i} )-A=\cup\left(X_{i}-A\right)\);
(v) \(\cap X_{i} \cup \cap Y_{j}=\cap_{i, j}\left(X_{i} \cup Y_{j}\right) ;\)
(vi) \(\cup X_{i} \cap \cup Y_{j}=\cup_{i, j}\left(X_{i} \cap Y_{j}\right)\).
Prove that
(i) \(\left(\cup A_{i}\right) \times B=\bigcup\left(A_{i} \times B\right)\);
(ii) \(\left(\cap A_{i}\right) \times B=\cap\left(A_{i} \times B\right)\);
(iii) \(\left(\cap_{i} A_{i}\right) \times\left(\cap_{j} B_{j}\right)=\bigcap_{i, j}\left(A_{i} \times B_{i}\right)\);
(iv) \(\left(\cup_{i} A_{i}\right) \times\left(\bigcup_{j} B_{j}\right)=\bigcup_{i, j}\left(A_{i} \times B_{j}\right)\).