SHOW WORK AND EXPLAIN! If P = {Bj| j J} is a partition of set S and each block B

Question

SHOW WORK AND EXPLAIN!

If P = {Bj| j J} is a partition of set S and each block Bj of this partition is further partitioned into = {C1 |i Ij}, then the collection = of the blocks of the blocks of the partition P forms a partition of the set S. and we say partition p of 5 is finer than partition p or is a refinement of or partition is coarser than partition . Two partitions b and D of a set S are given. Determine whether is finer than D, D is finer than b, or neither. S = [0. l]; b = {|0, 1 / 2), [1 / 2, 1]}, D = {[0, 1 / 2), [1 / 2, 3 / 4). [3 / 4, 1]} S = N; b = {{even natural numbers}, {odd natural numbers}}, D = {{even natural numbers 99}. {odd natural numbers}. S = R; b = {Q, R Q}, D = {(- infinity, 0). [0, infinity)}. S = R; b = {Q, R Q}, D = {{x)|x R}.

Explanation / Answer

labelling the first partition as A and the second as B, a) B is finer than A as the set [1/2,1] in A is further broken down in the sets [1/2,3/4], [3/4,1] in B b) Again B is finer than A as A contains only the two sets: odd, even whereas B contains the sets odd, even99, which further breaks down the even set into two finer sets. c) Neither. Neither of Q or R/Q is broken down into smaller sets, same with (-inf,0), [0,inf] d) B is finer than A as B contains the individual real numbers as sets {x}. If we club the rationals we can get Q and if we club the irrationals, we get R/Q, which are members of A. Thus B is finer than A. Hope this helps.

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