38. Let 2 = R0, the space of all sequences of the form w = (x1, x2, ...) (**) %3! where x; e R. Let o be a permutation of 1,..., n; that is, o is a 1-1 and onto map of (1, ...,n) (1,...,n}. If w is the sequence defined in (**), define ow to be the new sequence XoU), if j n. (ow)j = A finite permutation is of the form o for some n; that is, it juggles a finite initial segment of all positive integers. A set A CN is permutable if A =oA := (ow : w E A} for all finite permutations o. (i) Let Bn, n 2 1 be a sequence of subsets of R. Show that {w = (x1, x2, ...):*i € B, i.o. } and {w = (x1, x2, ...):V xị € B, i.o.) i=1 are permutable. (ii) Show the permutable sets form a a-field.
38. Let 2 = R0, the space of all sequences of the form w = (x1, x2, ...) (**) %3! where x; e R. Let o be a permutation of 1,..., n; that is, o is a 1-1 and onto map of (1, ...,n) (1,...,n}. If w is the sequence defined in (**), define ow to be the new sequence XoU), if j n. (ow)j = A finite permutation is of the form o for some n; that is, it juggles a finite initial segment of all positive integers. A set A CN is permutable if A =oA := (ow : w E A} for all finite permutations o. (i) Let Bn, n 2 1 be a sequence of subsets of R. Show that {w = (x1, x2, ...):*i € B, i.o. } and {w = (x1, x2, ...):V xị € B, i.o.) i=1 are permutable. (ii) Show the permutable sets form a a-field.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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38. Let 2 = R0, the space of all sequences of the form
w = (x1, x2, ...)
(**)
where x; e R. Let o be a permutation of 1,..., n; that is, o is a l-1 and
onto map of (1, ..., n) + (1,..., n}. If w is the sequence defined in (**),
define ow to be the new sequence
XoU), if j sn,
if j > n.
(ow)j =
A finite permutation is of the form o for some n; that is, it juggles a finite
initial segment of all positive integers. A set A CN is permutable if
A = oA := (ow : w e A)
for all finite permutations o.
(i) Let Br, n2 1 be a sequence of subsets of R. Show that
{w = (x1, x2, ...):x; e B, i.o. }
and
{w = (x1, x2,...): V xi
e B, i.o.)
i=1
are permutable.
(ii) Show the permutable sets form a o-field.
20 Ror a enhent ACNof non-negative integers, write card A) for the number
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