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Mathematics LibreTexts

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

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 1.2.E.1

For the relations specified in Problem 7 of §§1-3, find DR,DR, and R1. Also, find R[A] and R1[A] if
 (a) A={12}; (b) A={1} (c) A={0}; (d) A= ;  (e) A={0,3,15}; (f) A={3,4,7,0,1,6} (g) A={x|20<x<5}

Exercise 1.2.E.2

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

Exercise 1.2.E.3

Prove that
(i) R[AB]=R[A]R[B];
(ii) R[AB]R[A]R[B];
(iii) R[AB]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 {Ai|iI}.

Exercise 1.2.E.4

Under which conditions are the following statements true?
(i) R[x]=;(ii) R1[x]=;(iii) R[A]=;(iv) R1[A]=;

Exercise 1.2.E.5

Let f:NN(N={ naturals }). For each of the following functions, specify f[N], i.e., Df, and determine whether f is one to one and onto N, given that for all xN,
 (i) f(x)=x3; (ii) f(x)=1; (iii) f(x)=|x|+3 (iv) f(x)=x2;(v)f(x)=4x+5
Do all this also if N denotes
(a) the set of all integers;
(b) the set of all reals.

Exercise 1.2.E.6

Prove that for any mapping f and any sets A,B,Ai(iI),
(a) f1[AB]=f1[A]f1[B];
(b) f1[AB]=f1[A]f1[B];
(c) f1[AB]=f1[A]f1[B];
(d) f1[iAi]=if1[Ai];
(e) f1[iAi]=if1[Ai].
Compare with Problem 3.
[Hint: First verify that xf1[A] iff xDf and f(x)A.]

Exercise 1.2.E.7

Let f be a map. Prove that
(a) f[f1[A]]A;
(b) f[f1[A]]=A if ADf;
(c) if ADf and f is one to one, A=f1[f[A]]/
Is f[A]Bf[Af1[B]]?

Exercise 1.2.E.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)|xy is divisible by a fixed n};
(b) R={(x,y)|xy is odd };
(c) R={(x,y)|xy is a prime }.
(x,y,n denote integers.) 

Exercise 1.2.E.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.


1.2.E: Problems on Relations and Mappings (Exercises) is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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