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

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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.