1.1.E: Problems in Set Theory (Exercises)

Exercise \(\PageIndex{1}\)

Prove Theorem 1 (show that \(x\) is in the left-hand set iff it is in the right-hand set). For example, for \((\mathrm{d}),\)
\[
\begin{aligned} x \in(A \cup B) \cap C & \Longleftrightarrow[x \in(A \cup B) \text { and } x \in C] \\ & \Longleftrightarrow[(x \in A \text { or } x \in B), \text { and } x \in C] \\ & \Longleftrightarrow[(x \in A, x \in C) \text { or }(x \in B, x \in C)]. \end{aligned}
\]

Exercise \(\PageIndex{2}\)

Prove that
(i) \(-(-A)=A\);
(ii) \(A \subseteq B\) iff \(-B \subseteq-A\).

Exercise \(\PageIndex{3}\)

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.

Exercise \(\PageIndex{4}\)

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

Exercise \(\PageIndex{5}\)

Describe geometrically the following sets on the real line:
\[
\begin{array}{ll}{\text { (i) }\{x | x<0\} ;} & {\text { (ii) }\{x| | x |<1\}}; \\ {\text { (iii) }\{x| | x-a |<\varepsilon\} ;} & {\text { (iv) }\{x | a<x \leq b\}}; \\ {\text { (v) }\{x| | x |<0\}}. \end{array}
\]

Exercise \(\PageIndex{6}\)

Let \((a, b)\) denote the set
\[
\{\{a\},\{a, b\}\}
\]
(Kuratowski's definition of an ordered pair).
(i) Which of the following statements are true?
\[
\begin{array}{ll}{\text { (a) } a \in(a, b) ;} & {\text { (b) }\{a\} \in(a, b)}; \\ {\text { (c) }(a, a)=\{a\} ;} & {\text { (d) } b \in(a, b)}; \\ {\text { (e) }\{b\} \in(a, b) ;} & {\text { (f) }\{a, b\} \in(a, b)}. \end{array}
\]
(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 \(\{a\} \neq a . ]\)

Exercise \(\PageIndex{7}\)

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) \(\left\{(x, y) | x^{2}-2 x y+y^{2}=0\right\}\).

Exercise \(\PageIndex{8}\)

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. \(]\)

Exercise \(\PageIndex{9}\)

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) \(\cup X_{i} \cap \cup Y_{j}=\cup_{i, j}\left(X_{i} \cap Y_{j}\right)\).

Exercise \(\PageIndex{10}\)

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) \(\left(\cup_{i} A_{i}\right) \times\left(\bigcup_{j} B_{j}\right)=\bigcup_{i, j}\left(A_{i} \times B_{j}\right)\).