1.2.E: Problems on Relations and Mappings (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
For the relations specified in Problem 7 of §§1-3, find DR,D′R, and R−1. Also, find R[A] and R−1[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}
Prove that if A⊆B, then R[A]⊆R[B]. Disprove the converse by a counterexample.
Prove that
(i) R[A∪B]=R[A]∪R[B];
(ii) R[A∩B]⊆R[A]∩R[B];
(iii) R[A−B]⊇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|i∈I}.
Under which conditions are the following statements true?
(i) R[x]=∅;(ii) R−1[x]=∅;(iii) R[A]=∅;(iv) R−1[A]=∅;
Let f:N→N(N={ naturals }). For each of the following functions, specify f[N], i.e., D′f, and determine whether f is one to one and onto N, given that for all x∈N,
(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.
Prove that for any mapping f and any sets A,B,Ai(i∈I),
(a) f−1[A∪B]=f−1[A]∪f−1[B];
(b) f−1[A∩B]=f−1[A]∩f−1[B];
(c) f−1[A−B]=f−1[A]−f−1[B];
(d) f−1[⋃iAi]=⋃if−1[Ai];
(e) f−1[⋂iAi]=⋂if−1[Ai].
Compare with Problem 3.
[Hint: First verify that x∈f−1[A] iff x∈Df and f(x)∈A.]
Let f be a map. Prove that
(a) f[f−1[A]]⊆A;
(b) f[f−1[A]]=A if A⊆D′f;
(c) if A⊆Df and f is one to one, A=f−1[f[A]]/
Is f[A]∩B⊆f[A∩f−1[B]]?
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 is divisible by a fixed n};
(b) R={(x,y)|x−y is odd };
(c) R={(x,y)|x−y is a prime }.
(x,y,n 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.