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Q3 For a set A, let PA be the set of partitions of A. A partition R= {R1, R2,.., Rn} is a refinement of a partition P= {P1,P2,..., Pm} if for every R, E R, there exists P, e P such that R, C P;. For example, for the set A = {1,2, 3, 4, 5}, we consider the three partitions W = {{1,2},{4}, {3,5}}, X = {{1,2}, {3,4,5}}, and Y= {{1,2,3},{4,5}}. %3D Now W is a refinement of X, but W is not a refinement of Y. Let R be a relation on Pa that satisfies RRP if and only if R is a refinement of P. (a) Prove that for any set A, R is a partial order on PA.

Q3 For a set A, let PA be the set of partitions of A. A partition R= {R1, R2,.., Rn} is a refinement of a partition P= {P1,P2,..., Pm} if for every R, E R, there exists P, e P such that R, C P;. For example, for the set A = {1,2, 3, 4, 5}, we consider the three partitions W = {{1,2},{4}, {3,5}}, X = {{1,2}, {3,4,5}}, and Y= {{1,2,3},{4,5}}. %3D Now W is a refinement of X, but W is not a refinement of Y. Let R be a relation on Pa that satisfies RRP if and only if R is a refinement of P. (a) Prove that for any set A, R is a partial order on PA.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.1: Definition Of A Group
Problem 43E: Write out the elements of P(A) for the set A={ a,b,c }, and construct an addition table for P(A)...
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Q3 For a set A, let PA be the set of partitions of A. A partition R= {R1, R2,..., Rn} is a refinement of a partition P = {P, P2,..., Pm} if for every R; eR, there exists P; e P such that R C P,. For example, for the set A = {1,2, 3, 4, 5}, we consider the three partitions W = {{1,2}, {4}, {3, 5}}, X = {{1,2}, {3,4, 5}}, and Y = {{1,2,3}, {4,5}}. Now W is a refinement of X, but W is not a refinement of Y. Let R be a relation on Pa that satisfies RRP if and only if R is a refinement of P. (a) Prove that for any set A, R is a partial order on PA. (b) Give the values of the Stirling numbers S(4, 1), S(4,2), S(4, 3) and S(4,4). How many distinct partitions are there of a set with 4 elements? (c) If A = {1,2, 3,4}, draw the Hasse diagram for the partial order R on PA. For part (c), you may use a condensed notation to avoid a lot of set brackets by listing element with a dash to indicate a new part. For the examples above, we would write W as 12-4-35, X as 12-345 and Y as 123-45. + Drag and drop an image or PDF file or click to browse...
Transcribed Image Text:Q3 For a set A, let PA be the set of partitions of A. A partition R= {R1, R2,..., Rn} is a refinement of a partition P = {P, P2,..., Pm} if for every R; eR, there exists P; e P such that R C P,. For example, for the set A = {1,2, 3, 4, 5}, we consider the three partitions W = {{1,2}, {4}, {3, 5}}, X = {{1,2}, {3,4, 5}}, and Y = {{1,2,3}, {4,5}}. Now W is a refinement of X, but W is not a refinement of Y. Let R be a relation on Pa that satisfies RRP if and only if R is a refinement of P. (a) Prove that for any set A, R is a partial order on PA. (b) Give the values of the Stirling numbers S(4, 1), S(4,2), S(4, 3) and S(4,4). How many distinct partitions are there of a set with 4 elements? (c) If A = {1,2, 3,4}, draw the Hasse diagram for the partial order R on PA. For part (c), you may use a condensed notation to avoid a lot of set brackets by listing element with a dash to indicate a new part. For the examples above, we would write W as 12-4-35, X as 12-345 and Y as 123-45. + Drag and drop an image or PDF file or click to browse...
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