# Set theory, inverse difference?

I need to find the name and symbol for difference/complement, but the other way around?

to illustrate:

G1 = {a,b,c}
M = {b,c,d}

G1 ∪ M = {a,b,c,d}  #union
G1 ∩ M = {b,c}      #intersection
G1 \ M = {a}        #difference
G1 ∆ M = {a,d}      #symmetric difference
G1 ? M = {d}        # ????


If I understand correctly, then usually you'd just do M/G1, however this is for use in a stateful protocol where G1 is the current state, M is the message being sent along with the operation (one of the above) to be executed on it to change the state to n+1.

So what's ? in the above?

Many thanks in advance, and hope I don't sound too dim!

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## 3 Answers

I have never seen an explicit symbol $\ast$ such that for two sets: $$A\ast B=B\setminus A$$

In mathematics we would simply write $B\setminus A$. However $\setminus$ is such that the order of the variables matters (most of the time anyway). If you want a symbol as $\ast$ is described above, you would have to define one yourself.

If you wish to define this symbol for a specific aim (e.g. a stateful protocol) then it is perfectly legitimate to write a definition for such symbol. For example:

If $A$ and $B$ are two sets, we will write $A\boxplus B=B\setminus A=\{b\in B\mid b\notin A\}$.

It may be that such symbol was defined before by someone. However not all notation catches on in mathematics, and perhaps this operation was defined before but was abandoned for one reason or another.

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Perfect, thank you :) – nathan Jun 6 '12 at 15:13

Can you use set complements? If yes:

$$M \setminus G_1 = M \cap G_1^\complement = G_1^\complement \cap M$$

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... = {d}, in the OP's example. – Zach N Jun 5 '12 at 19:56

I don't know the limits you have to deal with - are you allowed to repeat the operations? If that is the case you might consider

# $$(G_1\Delta M)\setminus(G_1\setminus M)$$

In words: $(G_1\Delta M)$ is what is not in both minus what is only in $G_1$.

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