1.2.E: Problems on Relations and Mappings (Exercises)
For the relations specified in Problem 7 of §§1-3, find \(D_{R}, D_{R}^{\prime},\) and \(R^{-1}\). Also, find \(R[A]\) and \(R^{-1}[A]\) if
\[
\begin{array}{ll}{\text { (a) } A=\left\{\frac{1}{2}\right\} ;} & {\text { (b) } A=\{1\}} \\ {\text { (c) } A=\{0\} ;} & {\text { (d) } A=\emptyset \text { ; }} \\ {\text { (e) } A=\{0,3,-15\} ;} & {\text { (f) } A=\{3,4,7,0,-1,6\}} \\ {\text { (g) } A=\{x |-20<x<5\}}\end{array}
\]
Prove that if \(A \subseteq B\), then \(R[A] \subseteq R[B] .\) Disprove the converse by a counterexample.
Prove that
(i) \(R[A \cup B]=R[A] \cup R[B]\);
(ii) \(R[A \cap B] \subseteq R[A] \cap R[B]\);
(iii) \(R[A-B] \supseteq R[A]-R[B]\).
Disprove reverse inclusions in (ii) and (iii) by examples. Do (i) and (ii) with \(A, B\) replaced by an arbitrary set family \(\left\{A_{i} | i \in I\right\}\).
Under which conditions are the following statements true?
\[
\begin{array}{cc} \text{(i) } R[x] = \emptyset ; & \text{(ii) } R^{-1}[x] = \emptyset ; \\ \text{(iii) } R[A] = \emptyset ; & \text{(iv) } R^{-1}[A] = \emptyset ; \end{array}
\]
Let \(f : N \rightarrow N(N=\{\text { naturals }\}) .\) 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 \(x \in N\),
\[
\begin{array}{ll}{\text { (i) } f(x)=x^{3} ;} & {\text { (ii) } f(x)=1 ; \quad \text { (iii) } f(x)=|x|+3} \\ {\text { (iv) } f(x)=x^{2} ;} & {(\mathrm{v}) f(x)=4 x+5}\end{array}
\]
Do all this also if \(N\) denotes
(a) the set of all integers;
(b) the set of all reals.
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 \(f(x) \in A . ]\)
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 \(f[A] \cap B \subseteq f\left[A \cap f^{-1}[B]\right] ?\)
Is \(R\) an equivalence relation on the set \(J\) of all integers, and, if so, what are the \(R\) -classes, if
(a) \(R=\{(x, y) | x-y \text { is divisible by a fixed } n\}\);
(b) \(R=\{(x, y) | x-y \text { is odd }\}\);
(c) \(R=\{(x, y) | x-y \text { is a prime }\}\).
\((x, y, n \text { denote integers.) }\)
Is any relation in Problem 7 of §§1-3 reflexive? Symmetric? Transitive?
10. Show by examples that \(R\) may be
(a) reflexive and symmetric, without being transitive;
(b) reflexive and transitive without being symmetric.
Does symmetry plus transitivity imply reflexivity? Give a proof or counterexample.