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Taking U to be the universe of discourse, let R be the set of subsets of U and define the operations +, *, for R by

$A + B \equiv A \cup B - A \cap B$

$A \circ B \equiv A \cap B$

for subsets A, B of U. Define $0 \equiv \emptyset$ and $e \equiv U$. Then $(R,0,e,+,\circ)$ is a ring.

Compute the solutions X to the equations below using only $A, B, +, \circ$

i. $A + X = 0$

ii. $A + X = B$

iii. $X = (A-B)(A+B)$

I understand that X in number i. should be A, and I found in ii. that X can be written as $B - A \cup (B' \cap A)$, but I can't write it in terms of $A,B,+,\circ$ like the question asks.

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    Hint: This is actually isomorphic to the ring of functions $U\to \mathbb F_2$ under pointwise operations in $\mathbb F_2$. (Every set maps to its indicator function). Since $A+A=0$ for all $A$, it has characteristic 2, and therefore subtraction _in the ring_ is the same as the addition. (It would be notationally easier if you used $\setminus$ instead of $-$ for set difference, such that you could write subtraction in the ring with standard notation).2012-04-23
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    Oh man, thank you so much. I just had to realize A + A = A - A2012-04-23

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