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Question
Given  a nonempty set S, we select a fixed subset S0⊆S and use it to define two functions on the set of all subsets of S:
f(A)=A∪S0
g(A)=A\S0

Given  a nonempty set S, we select a fixed subset S0⊆S and use it to define two functions on the set of all subsets of S:

f(A)=A∪S0
g(A)=A\S0

For what choices of S0 do these functions commute?
Only for S0=∅
or
Only for S0=S
or
For all S0 ?

Answer
I assume you are denoting the set of all subsets of S (the power set of S) as A. Also, your notation A\S0 is new to me so I'm going to assume you mean A/S0 = quotient group {a*S0|a∈A}, which is defined only if A is a group and S0 is a normal subgroup of A.  Lastly, by commuting I assume you mean fοg = gοf where ο means composition.

Since S0 is a set of S then it is a set of A so that f(A) = A∪S0 = A. Then gοf(A) = g(A) = A/S0 = left cosets of S0.

fοg(A) = f(A/S0) = (A/S0)∪S0 = A/S0 since A contains the identity element, e, so that eS0 = S0 is one of the cosets. So the functions would seem to commute for all S0. However, it is not guaranteed that A is a group an arbitrary set S.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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