7.3.E: Problems on Set Families
- Last updated
- Jan 15, 2020
- Save as PDF
- Page ID
- 32345
( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercise 7.3.E.1
1. Verify Examples (a),(b), and (c).
Exercise 7.3.E.1′
Prove Theorem 1 for rings.
Exercise 7.3.E.2
Show that in Definition 1 "∅∈M" may be replaced by "M≠∅."
[Hint: ∅=A−A.]
Exercise 7.3.E.3
⇒ Prove that M is a field (σ-field if M≠∅,M is closed under finite (countable) unions, and
(∀A∈M)−A∈M.
[Hint: A−B=−(−A∪B);S=−∅.]
Exercise 7.3.E.4
Prove Theorem 2 for set fields.
Exercise 7.3.E.∗4′
Does Note 1 apply to semirings?
Exercise 7.3.E.5
Prove Note 2.
Exercise 7.3.E.5′
Prove Theorem 3 in detail.
Exercise 7.3.E.6
Prove Theorem 4 and show that the product M˙×N of two rings need not be a ring.
[Hint: Let S=E1 and M=N=2S. Take A,B as in Theorem 1 of §1. Verify that A−B∉M,M˙×N.]
Exercise 7.3.E.7
⇒ Let R,R′ be the rings (σ-rings, fields, σ-fields) generated by M and N, respectively. Prove the following.
(i) If M⊆N, then R⊆R′.
(ii) If M⊆N⊆R, then R=R′.
(iii) If
M={open intervals in En}
and
N={all open sets in En},
then R=R′.
[Hint: Use Lemma 2 in §2 for (iii). Use the minimality of R and R′.]
Exercise 7.3.E.8
Is any of the following a semiring, ring, σ-ring, field, or σ-field? Why?
(a) All infinite intervals in E1.
(b) All open sets in a metric space (S,ρ).
(c) All closed sets in (S,ρ).
(d) All "clopen" sets in (S,ρ).
(e) {X∈2S|−X finite}.
(f) {X∈2S|−X countable}.
Exercise 7.3.E.9
⇒ Prove that for any sequence {An} in a ring R, there is
(a) an expanding sequence {Bn}⊆R such that
(∀n)Bn⊇An
and
⋃nBn=⋃nAn; and
(b) a contracting sequence Cn⊆An, with
⋂nCn=⋂nAn.
(The latter holds in semirings, too.)
[Hint: Set Bn=⋃n1Ak,Cn=⋂n1Ak.]
Exercise 7.3.E.10
⇒ The symmetric difference, A△B, of two sets is defined
A△B=(A−B)∪(B−A).
Inductively, we also set
△1k=1Ak=A1
and
△n+1k=1Ak=(△nk=1Ak)△An+1.
Show that symmetric differences
(i) are commutative,
(ii) are associative, and
(iii) satisfy the distributive law:
(A△B)∩C=(A∩C)△(B∩C).
[Hint for (ii): Set A′=−A,A−B=A∩B′. Expand (A△B)△C into an expression symmetric with respect to A,B, and C.]
Exercise 7.3.E.11
Prove that M is a ring iff
(i) ∅∈M;
(ii) (∀A,B∈M)A△B∈M and A∩B∈M (see Problem 10); equivalently,
(ii') A△B∈M and A∪B∈M.
[Hint: Verify that
A∪B=(A△B)△(A∩B)
and
A−B=(A∪B)△B,
while
A∩B=(A∪B)△(A△B).]
Exercise 7.3.E.12
Show that a set family M≠∅ is a σ-ring iff one of the following conditions holds.
(a) M is closed under countable unions and proper differences (X−Y with X⊇Y);
(b) M is closed under countable disjoint unions, proper differences, and finite intersections; or
(c) M is closed under countable unions and symmetric differences (see Problem 10).
[Hints: (a) X−Y=(X∪Y)−Y, a proper difference.
(b) X−Y=X−(X∩Y) reduces any difference to a proper one; then
X∪Y=(X−Y)∪(Y−X)∪(X∩Y)
shows that M is closed under all finite unions; so M is a ring. Now use Corollary 1 in §1 for countable unions.
(c) Use Problem 11.]
Exercise 7.3.E.13
From Problem 10, treating △ as addition and ∩ as multiplication, show that any set ring M is an algebraic ring with unity, i.e., satisfies the six field axioms (Chapter 2, §§1-4), except V(b) (existence of multiplicative inverses).
Exercise 7.3.E.14
A set family H is said to be hereditary iff
(∀X∈H)(∀Y⊆X)Y∈H.
Prove the following.
(a) For every family M⊆2S, there is a "smallest" hereditary ring H⊇M (H is said to be generated by M). Similarly for σ-rings, fields, and σ-fields.
(b) The hereditary σ-ring generated by M consists of those sets which can be covered by countably many M-sets.
Exercise 7.3.E.15
Prove that the field (σ-field in S, generated by a ring (σ-ring R, consists exactly of all R-sets and their complements in S.
Exercise 7.3.E.16
Show that the ring R generated by a set family C≠∅ consists of all sets of the form
△nk=1Ak
(see Problem 10), where each Ak∈Cd (finite intersection of C-sets).
[Outline: By Problem 11, R must contain the family (call it M) of all such △nk=1Ak. (Why?) It remains to show that M is a ring ⊇C.
Write A+B for A△B and AB for A∩B; so each M-set is a "sum" of finitely many "products"
A1A2⋯An.
By algebra, the "sum" and "product" of two such "polynomials" is such a polynomial itself. Thus
(∀X,Y∈M)X△Y and X∩Y∈M.
Now use Problem 11.]
Exercise 7.3.E.17
AUse Problem 16 to obtain a new proof of Theorem 2 in §1 and Corollary 2 in the present section.
[Hints: For semirings, C=Cd. (Why?) Thus in Problem 16, Ak∈C.
Also,
(∀A,B∈C)A△B=(A−B)∪(B−A)
where A−B and B−A are finite disjoint unions of C-sets. (Why?)
Deduce that A△B∈C′s and, by induction,
△nk=1Ak∈C′s;
so R⊆C′s⊆R. (Why?)]
Exercise 7.3.E.18
Given a set A and a set family M, let
\[A \cap{\dot\} \mathcal{M}\]
be the family of all sets A∩X, with X∈M; similarly,
N˙∪(M˙−A)={ all sets Y∪(X−A), with Y∈N,X∈M}, etc.
Show that if M generates the ring R, then \boldsymbol{A \cap{\dot} \mathcal{M}} generates the ring
\boldsymbol{\mathcal{R}^{\prime}=A \cap{\dot} \mathcal{R}.}
Similarly for σ-rings, fields, σ-fields.
[Hint for rings: Prove the following.
(i) A∩R is a ring.
(ii) M⊆R′∪(R±A), with R′ as above.
(iii) R∪(R÷A) is a ring (call it N).
(iv) By (ii), \mathcal{R} \subseteq \mathcal{N}, so A \cap \mathcal{R} \subseteq A \cap \mathcal{N} \subseteq \mathcal{R}^{\prime}\right.
(v) A \cap \mathcal{R} \supseteq \mathcal{R}^{\prime}(\text { for } A \cap \mathcal{R} \supseteq A \cap \mathcal{M}).
Hence \mathcal{R}^{\prime}=A \cap \mathcal{R}.]