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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 ?

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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