Prove that $+$ is associative where $A+B=( A \setminus B )\cup( B \setminus A ) $.

Question

If $A$ and $B$ are sets, then $A+B=( A \setminus B )\cup( B \setminus A ) $.

Prove that $+$ is an associative operation.

How do I prove these?

By the way, the symmetric difference is usually written $\oplus$, the TeX for which is \oplus. — Alexander Gruber, Mar 22 '13 at 19:15
@AlexanderGruber I've never seen $\oplus$ for symmetric difference. Only $\triangle$ and $\ominus$. — kahen, Mar 22 '13 at 19:18
@AlexanderGruber I think in this question it's meant that one defines some operation "$+$" by basic set operations and then one has to prove that it's associative. And BTW, it's not $\oplus$. — Kaster, Mar 22 '13 at 19:19
@Kaster Yes, that is obvious. My comment was to inform OP of an alternative notation for the operation. — Alexander Gruber, Mar 22 '13 at 19:24
@AlexanderGruber I get it, I and kahen are just saying that alternative notation you provided is not for symmetric difference. — Kaster, Mar 22 '13 at 19:25
And I have seen all three of these symbols used in the literature for symmetric difference: $\oplus, \ominus,$ and $\Delta$. — Alexander Gruber, Mar 22 '13 at 19:25
What can I say? I have seen $\oplus$, usually in group theory. It comes from the XOR usage. I don't object to $\ominus$ or $\bigtriangleup$. — Alexander Gruber, Mar 22 '13 at 19:28
See Symmetric Difference is Associative at ProofWiki and Another proof of $A \triangle (B \triangle C)=(A \triangle B) \triangle C$. Somewhat related also: Symmetric Difference Identity and Symmetric Difference with Intersection forms Ring at ProofWiki. — Martin Sleziak, Mar 22 '13 at 21:05

score 4 · Answer 1 · answered Mar 22 '13 at 19:31

4

Hint The characteristic function of the set $A\subset X$ is defined by $$\begin{array}{ccc} 1_A: & X \rightarrow& \{0,1\} \\ & x\mapsto & \left\{ \begin{array}{ll} 1 & \hbox{if}\, x\in A \\ 0 & \hbox{if}\, x\notin A \end{array} \right. \end{array}$$ then it's easy to prove that $$1_{A\cap B}=1_A 1_B\quad;\quad 1_{A^c}=1-1_A\quad;\quad1_A=1_B\iff A=B$$ now it's easy to show that $$ A+(B+C)=(A+B)+C$$

answered Mar 22 '13 at 19:31

is ther any law for $1_ {(A∪B)}$ ? – FNH Mar 22 '13 at 20:27
is the relation , $$ \textbf{1}{B} + \textbf{1}{C} = \textbf{1}_{B \Delta C} $$ true ? where ${B \Delta C}$ is the symmetric diffrence – FNH Mar 22 '13 at 20:48
@MrWhy If $A$ and $B$ are disjoint subset then $1_{A\cup B}=1_A+1_B$, and since $B\Delta C=(B\cap C^c)\cup (C\cap B^c)$ then apply the relations given in my answer an you find $1_{B\Delta C}=1_B+1_C-21_B1_C$. – Mar 22 '13 at 21:43
according to wikipedia , http://en.wikipedia.org/wiki/Indicator_function , $1_{A∪B} $ = $ 1_{A} $ + $1_{B}$ - $ ( 1_{A} 1_{B} ) $ not
$1_{A∪B} $ = $ 1_{A} $ + $1_{B}$
– FNH Mar 22 '13 at 22:48
1

@MrWhy This is true in general case but in case of disjoint subsets we have $1_{A\cup B}=1_A+1_B$ as I said. – Mar 23 '13 at 08:43

score 3 · Accepted Answer · answered Mar 22 '13 at 19:06

3

One approach would be to first prove that $x \in A+B$ is equivalent to "x belongs to exactly one of the sets $A$ and $B$".

You can then deduce that $x \in (A+B)+C$ and $x \in A+(B+C)$ are both equivalent to the statement "$x$ belongs to either exactly one or all three of the sets $A, B, C$".

I will leave you to work out the details, but let us know if you get stuck.

answered Mar 22 '13 at 19:06

Old John

19,875

thanx ! this approach is Great ! i completed the details
but is there other approach using the laws of set theory ?

what is the laws which will be used in a proof like this ?
– FNH Mar 22 '13 at 19:26
It is possible (but very messy, I think) to prove the result using the fact that $A \setminus B = A \cap B'$ (where $B'$ is the complement of $B$) and then doing a lot of set algebra. Another approach uses indicator functions and some simple algabra. – Old John Mar 22 '13 at 19:32
@CтарыйДжон Damn, I was looking to different Venn diagram all this time. Apologies for trying to start fruitless discussion. I removed all my posts to save myself from future embarrassment. Thank you. – Kaster Mar 22 '13 at 20:51

Prove that $+$ is associative where $A+B=( A \setminus B )\cup( B \setminus A ) $.

2 Answers2