Let A be a nonempty set and fix the set B where B⊆A. Define the relation R on ℘(A), the power set of A, as follows: for all subsets X,Y of A, X R Y iff B∩X = B∩Y.
Suppose we make some sets...
A={1,2,3}
B={1,2}
How do i find the partition of the power set of A? Do I need to know what X and Y are first?
well we can find that:
℘(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
therefore the partition of the power set of A would be:
[{},{3}],
[{1},{1,3}],
[{2},{2,3}],
[{1,2},{1,2,3}]
reasoning:
[{},{3}]
if X={}, Y={} and B={1,2}
B∩X = B∩Y = {} = {} which is true
if X={}, Y={3} and B={1,2}
B∩X = B∩Y = {} = {} which is true
if X={3}, Y={3} and B={1,2}
B∩X = B∩Y = {} = {} which is true
if X={3}, Y={} and B={1,2}
B∩X = B∩Y = {} = {} which is true
~~~~~~~~~~~~~~~~~~~~~~~~~~
[{1},{1,3}]
if X={1}, Y={1} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true
if X={1}, Y={1,3} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true
if X={1,3}, Y={1} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true
if X={1,3}, Y={1,3} and B={1,2}
B∩X = B∩Y = {1} = {1} which is true
~~~~~~~~~~~~~~~~~~~~~~~~~~
similarly for [{2},{2,3}] and [{1,2},{1,2,3}]