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Mathematics LibreTexts

7.3.E: Problems on Set Families

( \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: =AA.]

Exercise 7.3.E.3

Prove that M is a field (σ-field if M,M is closed under finite (countable) unions, and
(AM)AM.
[Hint: AB=(AB);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 ABM,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 MN, then RR.
(ii) If MNR, 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) {X2S|X finite}.
(f) {X2S|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)BnAn
and
nBn=nAn; and
(b) a contracting sequence CnAn, with
nCn=nAn.
(The latter holds in semirings, too.)
[Hint: Set Bn=n1Ak,Cn=n1Ak.]

Exercise 7.3.E.10

The symmetric difference, AB, of two sets is defined
AB=(AB)(BA).
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:
(AB)C=(AC)(BC).
[Hint for (ii): Set A=A,AB=AB. Expand (AB)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,BM)ABM and ABM (see Problem 10); equivalently,
(ii') ABM and ABM.
[Hint: Verify that
AB=(AB)(AB)
and
AB=(AB)B,
while
AB=(AB)(AB).]

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 (XY with XY);
(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) XY=(XY)Y, a proper difference.
(b) XY=X(XY) reduces any difference to a proper one; then
XY=(XY)(YX)(XY)
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
(XH)(YX)YH.
Prove the following.
(a) For every family M2S, there is a "smallest" hereditary ring HM (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 AkCd (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 AB and AB for AB; so each M-set is a "sum" of finitely many "products"
A1A2An.
By algebra, the "sum" and "product" of two such "polynomials" is such a polynomial itself. Thus
(X,YM)XY and XYM.
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, AkC.
Also,
(A,BC)AB=(AB)(BA)
where AB and BA are finite disjoint unions of C-sets. (Why?)
Deduce that ABCs and, by induction,
nk=1AkCs;
so RCsR. (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 AX, with XM; similarly,
N˙(M˙A)={ all sets Y(XA), with YN,XM}, 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) AR is a ring.
(ii) MR(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}.]


7.3.E: Problems on Set Families is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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