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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.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 48E: Let R be the set of all infinite sequences of real numbers, with the operations...
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we nue fro. From th ex era'se that U Cau we Say There for 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
Transcribed Image Text:we nue fro. From th ex era'se that U Cau we Say There for 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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