Prove that power set is a field?
1 Answer
The power set of a set is a commutative ring under the natural operations of union and intersection, but not a field under those operations, since it lacks inverse elements.
Explanation:
Given any set
This has natural operations of union
In more detail:

#2^S# is closed under#uu#
If#A, B in 2^S# then#A uu B in 2^S# 
There is an identity
#O/ in 2^S# for#uu#
If#A in 2^S# then#A uu O/ = O/ uu A = A# 
#uu# is associative
If#A, B, C in 2^S# then#A uu (B uu C) = (A uu B) uu C# 
#uu# is commutative
If#A, B in 2^S# then#A uu B = B uu A# 
#2^S# is closed under#nn#
If#A, B in 2^S# then#A nn B in 2^S# 
There is an identity
#S in 2^S# for#nn#
If#A in 2^S# then#A nn S = S nn A = A# 
#nn# is associative
If#A, B, C in 2^S# then#A nn (B nn C) = (A nn B) nn C# 
#nn# is commutative
If#A, B in 2^S# then#A nn B = B nn A# 
#nn# is left and right distributive over#uu#
If#A, B in 2^S# then#A nn (B uu C) = (A nn B) uu (A nn C)#
and#(A uu B) nn C = (A nn C) uu (B nn C)#
So
If
Otherwise note that