PlanetMath: symmetric difference on a finite number of sets

Recall that the symmetric difference operation on sets is commutative and associative. Therefore, one can speak of the symmetric difference of a finite collection of sets. More precisely, let $A_1,A_2,\ldots, A_n$ be sets, not necessarily pairwise distinct. The set $$A:=\displaystyle{\symsum{i=1}{n} A_i},$$ the symmetric difference of the sets $A_i$ , is well-defined.

Let $A,A_1,\ldots,A_n$ be defined as above. There is a curious property on $A$ :

From the two examples above, it seems that the approach to proving the proposition is to express the symmetric difference in CNF, and this is indeed the case.

To facilitate with the proof, let us introduce some notations. Start with sets $A_1,\ldots, A_n$ , which are assumed to be subsets of some set $U$ . Let $I$ and $C$ be the identity and complementation operations taking $A\subseteq U$ to $A$ and $A'$ respectively. Let $\mathbf{n}$ be the set $\lbrace 1,\ldots, n\rbrace$ . Let $F_n$ be the set of all functions from $\mathbf{n}$ into $\lbrace I,C\rbrace$ . For every $f\in F_n$ , we write $f_i$ for $f(i)$ . Finally, we partition $F_n$ into two sets $E_n$ and $O_n$ , where $E_n$ ($O_n$ ) consists of all functions $f$ such that $|f^{-1}(I)|$ is even (odd), respectively.

$$\displaystyle{\symsum{i=1}{n} A_i} = \bigcup_{f\in O_n} \bigcap_{i=1}^n f_i(A_i)$$

Proof. We prove this equation by induction on $n$ , for $n\ge 2$ . The case when $n=2$ is already discussed above. Now, $$\displaystyle{\symsum{i=1}{n+1} A_i} = \left(\displaystyle{\symsum{i=1}{n} A_i}\right) \Delta A_{n+1} = X\cup Y,$$ where $$X = \left(\displaystyle{\symsum{i=1}{n} A_i}\right)\cap A'_{n+1} \qquad \mbox{and} \qquad Y= A_{n+1} \cap \left( \displaystyle{\symsum{i=1}{n} A_i}\right)'.$$ By the induction hypothesis, $$\displaystyle{\symsum{i=1}{n} A_i} = \bigcup_{f\in O_n} \bigcap_{i=1}^n f_i(A_i),$$ so that $$X = \bigg( \bigcup_{f\in O_n} \bigcap_{i=1}^n f_i(A_i) \bigg) \cap A'_{n+1} = \bigcup_{f\in O_n} \bigcap_{i=1}^{n+1} \hat{f}_i(A_i), $$ where $\hat{f}:\mathbf{(n+1)}\to \lbrace I,C\rbrace$ is given by $\hat{f}_i=f_i$ if $i\in \mathbf{n}$ , and $\hat{f}_{n+1}=C$ .

Now, for any $x\in U$ , $x$ is either in an even number of $A_i$ 's, or an odd number of $A_i$ 's. This means that $$x\in \bigcup_{f\in E_n} \bigcap_{i=1}^n f_i(A_i) \qquad \mbox{or} \qquad x\in \bigcup_{f\in O_n} \bigcap_{i=1}^n f_i(A_i)$$ and never both. This shows that $U$ can be partitioned into the two sets above. In other words, $$\left( \bigcup_{f\in O_n} \bigcap_{i=1}^n f_i(A_i) \right)' = \bigcup_{f\in E_n} \bigcap_{i=1}^n f_i(A_i).$$ As a result, $$Y = A_{n+1} \cap \left( \bigcup_{f\in E_n} \bigcap_{i=1}^n f_i(A_i) \right)= \bigcup_{f\in E_n} \bigcap_{i=1}^{n+1} \overline{f}_i(A_i), $$ where $\overline{f}:\mathbf{(n+1)}\to \lbrace I,C\rbrace$ is given by $\overline{f}_i=f_i$ if $i\in \mathbf{n}$ , and $\hat{f}_{n+1}=I$ .

Every function $g\in O_{n+1}$ can be obtained from a function in $f\in F_n$ so that $g_i=f_i$ for $i\in \mathbf{n}$ . If $f\in O_n$ , then $g_{n+1}=C$ , and if $f\in E_n$ , then $g_{n+1}=I$ . This means that $O_{n+1}$ can be partitioned into two sets $O$ and $E$ , where $O$ (or $E$ ) contains all functions whose restriction to $\mathbf{n}$ are in $O_n$ (or $E_n$ ).

Therefore, \begin{eqnarray*} \bigcup_{g\in O_{n+1}} \bigcap_{i=1}^{n+1} g_i(A_i) &=& \bigg(\bigcup_{g\in O} \bigcap_{i=1}^{n+1} g_i(A_i)\bigg) \cup \bigg( \bigcup_{g\in E} \bigcap_{i=1}^{n+1} g_i(A_i)\bigg) \\ &=& \bigg(\bigcup_{f\in O_n} \bigcap_{i=1}^{n+1} \hat{f}_i(A_i)\bigg) \cup \bigg( \bigcup_{f\in E_n} \bigcap_{i=1}^{n+1} \overline{f}_i(A_i)\bigg) \\ &=& X\cup Y = \displaystyle{\symsum{i=1}{n+1} A_i}. \end{eqnarray*} $\qedsymbol$