1.2.E: Problems on Relations and Mappings (Exercises)

Exercise $\PageIndex{1}$

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
$$\begin{array}{ll}{\text { (a) } A=\left\{\frac{1}{2}\right\} ;} & {\text { (b) } A=\{1\}} \\ {\text { (c) } A=\{0\} ;} & {\text { (d) } A=\emptyset \text { ; }} \\ {\text { (e) } A=\{0,3,-15\} ;} & {\text { (f) } A=\{3,4,7,0,-1,6\}} \\ {\text { (g) } A=\{x |-20<x<5\}}\end{array}$$

Exercise $\PageIndex{2}$

Prove that if $A \subseteq B$, then $R[A] \subseteq R[B] .$ Disprove the converse by a counterexample.

Exercise $\PageIndex{3}$

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 $\left\{A_{i} | i \in I\right\}$.

Exercise $\PageIndex{4}$

Under which conditions are the following statements true?
$$\begin{array}{cc} \text{(i) } R[x] = \emptyset ; & \text{(ii) } R^{-1}[x] = \emptyset ; \\ \text{(iii) } R[A] = \emptyset ; & \text{(iv) } R^{-1}[A] = \emptyset ; \end{array}$$

Exercise $\PageIndex{5}$

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.

Exercise $\PageIndex{6}$

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 $f(x) \in A . ]$

Exercise $\PageIndex{7}$

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 $f[A] \cap B \subseteq f\left[A \cap f^{-1}[B]\right] ?$

Exercise $\PageIndex{8}$

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) $R=\{(x, y) | x-y \text { is a prime }\}$.
$(x, y, n \text { denote integers.) }$

Exercise $\PageIndex{9}$

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.