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# 1.1.E: Problems in Set Theory (Exercises)

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