MHB How Can I Show That the Symmetric Difference of Sets is Associative?

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To demonstrate that the symmetric difference of sets is associative, the discussion explores the expression \( (A \triangle B) \triangle C = A \triangle (B \triangle C) \). The approach involves manipulating set membership conditions and using characteristic functions to illustrate the equivalence. Participants suggest reducing both sides to disjunctive normal form or analyzing eight cases based on membership in sets A, B, and C. Ultimately, the conclusion is reached that the symmetric difference can be expressed in terms of set operations, confirming its associative property. This methodical breakdown affirms the validity of the associative law for symmetric differences.
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Hello! (Wave)I want to show that $ (A \triangle B) \triangle C=A \triangle (B \triangle C) $.

I have tried the following:

$$ x \in (A \triangle B) \triangle C \Leftrightarrow x \in (A \triangle B)\setminus C \lor x \in C \setminus (A \triangle B) \\ \Leftrightarrow (x \in A \triangle B \wedge x \notin C) \lor (x \in C \wedge x \notin A \triangle B ) \\ \Leftrightarrow (((x \in A \wedge x \notin B) \lor (x \in B \wedge x \notin A)) \wedge x \notin C)\\ \lor (x \in C \wedge (x \in A \cap B \lor x \notin A \cup B) ) \\ \Leftrightarrow ((x \in A \wedge x \notin B \wedge x \notin C) \lor (x \in B \wedge x \notin A \wedge x \notin C)) \lor (x \in C \wedge (x \in A \cap B \lor x \notin A \cup B) ) \\ \Leftrightarrow ((x \in A \wedge x \notin B \cup C) \lor (x \in B \wedge x \notin A \cup C))\lor (x \in C \wedge (x \in A \cap B \lor x \notin A \cup B) ) $$

How could we continue?
 
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If you are patient, it is possible to reduce both sides to a disjunctive normal form, i.e., a disjunction of conjunctions, where every element of a conjunction is $x\in S$ or $x\notin S$ and $S$ is one of $A$, $B$, or $C$. A similar approach is to consider eight cases where $x$ belongs or does not belong to each of $A$, $B$, and $C$, and for each of those cases calculate the truth value of both sides.

If you can rely on the fact that addition is associative in $\mathbb{Z}_2$, then you can notice that $x\in S_1\vartriangle S_2$ iff $\chi_1(x)+\chi_2(x)=1$ where $\chi_i(x)=\begin{cases}1,&x\in S_i\\0,&x\notin S_i\end{cases}$ is the characteristic function of $S_i$. Then
\[
\begin{align}
x\in (A\vartriangle B)\vartriangle C&\iff (\chi_A(x)+\chi_B(x))+\chi_C(x)=1\\
&\iff \chi_A(x)+(\chi_B(x)+\chi_C(x))=1\\&\iff x\in A\vartriangle (B\vartriangle C).
\end{align}
\]
 
evinda said:
Hello! (Wave)I want to show that $ (A \triangle B) \triangle C=A \triangle (B \triangle C) $.

I have tried the following:

$$ x \in (A \triangle B) \triangle C \Leftrightarrow x \in (A \triangle B)\setminus C \lor x \in C \setminus (A \triangle B) \\ \Leftrightarrow (x \in A \triangle B \wedge x \notin C) \lor (x \in C \wedge x \notin A \triangle B ) \\ \Leftrightarrow (((x \in A \wedge x \notin B) \lor (x \in B \wedge x \notin A)) \wedge x \notin C)\\ \lor (x \in C \wedge (x \in A \cap B \lor x \notin A \cup B) ) \\ \Leftrightarrow ((x \in A \wedge x \notin B \wedge x \notin C) \lor (x \in B \wedge x \notin A \wedge x \notin C)) \lor (x \in C \wedge (x \in A \cap B \lor x \notin A \cup B) ) \\ \Leftrightarrow ((x \in A \wedge x \notin B \cup C) \lor (x \in B \wedge x \notin A \cup C))\lor (x \in C \wedge (x \in A \cap B \lor x \notin A \cup B) ) $$

How could we continue?

$$(x\in A\wedge x\notin B\wedge x\notin C)\vee(x\in B \wedge x\notin A\wedge x\notin C)\vee[x\in C((x\in A\wedge x\in B)\vee(x\notin A\wedge x\notin B))]\Leftrightarrow(x\in A\wedge x\notin B\wedge x\notin C)\vee(x\in B \wedge x\notin A\wedge x\notin C)\vee(x\in C\wedge x\in A\wedge x\in B)\vee(x\in C\wedge x\notin A\wedge x\notin B)$$$$\Leftrightarrow[(x\in A\wedge x\notin B\wedge x\notin C)\vee(x\in C\wedge x\in A\wedge x\in B)]\vee[(x\in B \wedge x\notin A\wedge x\notin C)\vee(x\in C\wedge x\notin A\wedge x\notin B)]\Leftrightarrow [x\in A\wedge (( x\notin B\wedge x\notin C)\vee(x\in C\wedge x\in B))]\vee [x\notin A\wedge ((x\in B \wedge x\notin C)\vee(x\in C\wedge x\notin B))]$$

Can you go on from here??
 
Last edited:
solakis said:
$$(x\in A\wedge x\notin B\wedge x\notin C)\vee(x\in B \wedge x\notin A\wedge x\notin C)\vee[x\in C((x\in A\wedge x\in B)\vee(x\notin A\wedge x\notin B))]\Leftrightarrow(x\in A\wedge x\notin B\wedge x\notin C)\vee(x\in B \wedge x\notin A\wedge x\notin C)\vee(x\in C\wedge x\in A\wedge x\in B)\vee(x\in C\wedge x\notin A\wedge x\notin B)$$$$\Leftrightarrow[(x\in A\wedge x\notin B\wedge x\notin C)\vee(x\in C\wedge x\in A\wedge x\in B)]\vee[(x\in B \wedge x\notin A\wedge x\notin C)\vee(x\in C\wedge x\notin A\wedge x\notin B)]\Leftrightarrow [x\in A\wedge (( x\notin B\wedge x\notin C)\vee(x\in C\wedge x\in B))]\vee [x\notin A\wedge ((x\in B \wedge x\notin C)\vee(x\in C\wedge x\notin B))]$$

Can you go on from here??
In the above disjunction ,let us calculate each disjunct seperately.
So for :

$$[x\in A\wedge (( x\notin B\wedge x\notin C)\vee(x\in C\wedge x\in B))]$$ we have:

$$[x\in A\wedge (F\vee ( x\notin B\wedge x\notin C)\vee(x\in C\wedge x\in B)\vee F)]$$ (remember we are working in the domain of the algebra of propositions)
$$\Leftrightarrow x\in A\wedge[(x\in B\wedge x\notin B)\vee ( x\notin B\wedge x\notin C))\vee(x\in C\wedge x\in B)\vee (x\in C\wedge x\notin C)] $$

$$\Leftrightarrow x\in A\wedge[((x\in B\wedge x\notin B)\vee ( x\notin B\wedge x\notin C))\vee((x\in C\wedge x\in B)\vee (x\in C\wedge x\notin C))] $$in the $$((x\in B\wedge x\notin B)\vee ( x\notin B\wedge x\notin C))$$

get common factor $$x\notin B$$

in the $$((x\in C\wedge x\in B)\vee (x\in C\wedge x\notin C))$$

get common factor $$x\in C$$ and so we have:

$$\Leftrightarrow x\in A\wedge[x\notin B\wedge(x\in B\vee x\notin C)\vee x\in C\wedge(x\in B\vee x\notin C)]$$

$$\Leftrightarrow x\in A\wedge[(x\notin B\vee x\in C)\wedge (x\in B\vee x\notin C)] $$

And using D.Morgan we have:

$$\Leftrightarrow x\in A\wedge\neg[\neg(x\notin B\vee x\in C)\vee\neg (x\in B\vee x\notin C)] $$

$$\Leftrightarrow x\in A\wedge\neg[(x\in B\wedge x\notin C)\vee (x\notin B\wedge x\in C)] $$

$$\Leftrightarrow x\in A\wedge x\notin (B\triangle C)$$......1

And for the other disjunct $$[x\notin A\wedge ((x\in B \wedge x\notin C)\vee(x\in C\wedge x\notin B))]$$

We have:

$$\Leftrightarrow x\notin A\wedge x\in(B\triangle C)$$........2

And the disjunction of (1) and (2) is:

$$[x\in A\wedge x\notin (B\triangle C)]\vee[ x\notin A\wedge x\in(B\triangle C)]$$

$$\Leftrightarrow A\triangle(B\triangle C)$$
 

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