Problem 1
Prove that if \(A\) is countable but \(B\) is not, then \(B-A\) is uncountable.
Problem 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), $$ \begin{aligned} x \in(A \cup B) \cap C & \Longleftrightarrow[x \in(A \cup B) \text { and } x \in C] \\ & \Longleftrightarrow[(x \in A \text { or } x \in B), \text { and } x \in C] \\\ & \Longleftrightarrow[(x \in A, x \in C) \text { or }(x \in B, x \in C)]. \end{aligned} $$
Problem 2
Let \(f\) be a mapping, and \(A \subseteq D_{f} .\) Prove that (i) if \(A\) is countable, so is \(f[A]\); (ii) if \(f\) is one to one and \(A\) is uncountable, so is \(f[A]\).
Problem 2
Prove that if \(A \subseteq B,\) then \(R[A] \subseteq R[B] .\) Disprove the converse by a counterexample.
Problem 4
Show that between any real numbers \(a, b(a
Problem 5
Show that every infinite set \(A\) contains a countably infinite set, i.e., an infinite sequence of distinct terms.
Problem 5
Let \(f: N \rightarrow N(N=\\{\) naturals \(\\})\). For each of the following functions, specify \(f[N]\), i.e., \(D_{f}^{\prime},\) and determine whether \(f\) is one to one and onto \(N,\) given that for all \(x \in N\) (i) \(f(x)=x^{3} ;\) (ii) \(f(x)=1 ;\) (iii) \(f(x)=|x|+3 ;\) (iv) \(f(x)=x^{2}\) (v) \(f(x)=4 x+5\). Do all this also if \(N\) denotes (a) the set of all integers; (b) the set of all reals.
Problem 6
Prove that for any mapping \(f\) and any sets \(A, B, A_{i}(i \in I),\) (a) \(f^{-1}[A \cup B]=f^{-1}[A] \cup f^{-1}[B] ;\) (b) \(f^{-1}[A \cap B]=f^{-1}[A] \cap f^{-1}[B]\); (c) \(f^{-1}[A-B]=f^{-1}[A]-f^{-1}[B]\) (d) \(f^{-1}\left[\bigcup_{i} A_{i}\right]=\bigcup_{i} f^{-1}\left[A_{i}\right]\) (e) \(f^{-1}\left[\bigcap_{i} A_{i}\right]=\bigcap_{i} f^{-1}\left[A_{i}\right]\) Compare with Problem 3 . [Hint: First verify that \(x \in f^{-1}[A]\) iff \(x \in D_{f}\) and \(f(x) \in A\).]
Problem 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) \(a \in(a, b)\); (b) \(\\{a\\} \in(a, b)\); (c) \((a, a)=\\{a\\}\); (d) \(b \in(a, b)\); (e) \(\\{b\\} \in(a, b)\); \((\mathrm{f})\\{a, b\\} \in(a, b)\); (ii) Prove that \((a, b)=(u, v)\) iff \(a=u\) and \(b=v\). [Hint: Consider separately the two cases \(a=b\) and \(a \neq b,\) noting that \(\\{a, a\\}=\) \(\\{a\\} .\) Also note that \(\\{a\\} \neq a .]\)
Problem 7
Let \(f\) be a map. Prove that (a) \(f\left[f^{-1}[A]\right] \subseteq A\) (b) \(f\left[f^{-1}[A]\right]=A\) if \(A \subseteq D_{f}^{\prime}\) (c) if \(A \subseteq D_{f}\) and \(f\) is one to one, \(A=f^{-1}[f[A]]\). Is \(f[A] \cap B \subseteq f\left[A \cap f^{-1}[B]\right] ?\)
