Question

Please help me understand the following question and if it can be solved in written form please thank you so much

1. Let U be any set. Prove that for every B ∈ ℘(U) there is a unique D ∈ ℘(U) such that for every C ∈ ℘(U), C \ B = C ∩ D.

This problem is similar to Examples 3.6.2 and 3.6.4 and to Exercise 8 in Section 3.6 of your SNHU MAT299 textbook.

Answer 1

We are given a set U. Now, for any
BE PU
(i.e. for any subset B of U) there is a unique
DE P(U)
(another subset which is unique once B is given) such that
VC E P(U), C\B = CnD
.

Given B take
D=B
. Since
CB Cn B
, we have
CB Cn D
. This proves the existence of D.

To show uniqueness of D, suppose there is another set E which satisfies the same properties. Taking C=E we have
EB En D = EnE= E
. This implies
ECD
.Similarly if we take C=D, we get
DCE
.

Hence which proves the uniqueness of D.