Given 4 sets: $A, B, C, D$, Prove that if $A \triangle B \subseteq D, B \triangle C \subseteq D$, then $A \triangle C \subseteq D$

Question

Given four sets: $A, B, C, D$, suppose that $$A \triangle B \subseteq D \ \land \ B \triangle C \subseteq D $$ Prove: $A \triangle C \subseteq D$

[As $\triangle$ means symmetric difference]

My Attempt:

Notice that $A \triangle B = (A \setminus B) \cup (B \setminus A) = (A \cup B)\setminus (A\cap B)$

Therefore: $\left((A\cup B)\setminus(A\cap B)\right) \text{ and } \left((B\cup C)\setminus (B\cap C)\right) \subseteq D$

EDIT:

I'd like to prove: $(A \triangle C) \subseteq (A\triangle B)\cup(B\triangle C)$ $$\left((A\triangle B)\cup(B \triangle C)\right)=(A\cup B)\setminus(A\cap B)\cup(B\cup C)\setminus(B\cap C)$$ $$ =(A\cup B)\cup(B\cup C)\setminus(A\cap B)\cup(B\cap C)$$ $$=B\cup(A\cup C)\setminus B\cap(A\cup C)$$

$ \ $

Now -- can I say that $(A\cup C)\setminus(A\cap C) \subseteq (B\cup(A\cup C)\setminus B\cap(A\cup C)$?

my previous attempt:

Notice that $A \triangle B = (A \setminus B) \cup (B \setminus A) = (A \cup B)\setminus (A\cap B)$

Therefore: $\left((A\cup B)\setminus(A\cap B)\right) \text{ and } \left((B\cup C)\setminus (B\cap C)\right) \subseteq D$

I'll divide into cases:

I. let $x \in (A\triangle B) \land x\notin C \to x\in (A\setminus C) \subseteq (A \triangle C)$

Notice that $A\triangle B \subseteq D \to x\in D$, therefore $x \in A\triangle C \land x\in D$

II. let $x'\in (B \triangle C) \land \ x'\notin A \ $, then $x'\in (C\setminus A) \subseteq (A\triangle C)$ as $(B\triangle C) \subseteq D \to \ x'\in D$

III. Let $x'' \in (B\cap C) \land x\notin A$ Then $x\in (B\setminus A)\subseteq (A \triangle B) \subseteq D$, Hence $x'' \in D$

Notice that $x'' \in (C\setminus A) \subseteq (A \triangle C)$ , so $x'' \in D \land x''\in (A\triangle C)$.

I don't know if I covered all possible cases (and can't find a way to do so. using contradiction didn't helped). What can be done at such case?

Answer 1

The proof is possibly good, but once you know associativity of symmetric difference, you can realize that $$ (A\mathbin{\triangle}B)\mathbin{\triangle}(B\mathbin{\triangle}C)= A\mathbin{\triangle}(B\mathbin{\triangle}B)\mathbin{\triangle}C= A\mathbin{\triangle}\emptyset\mathbin{\triangle}C=A\mathbin{\triangle}C $$ Thus you only need to show that $X\subseteq D$ and $Y\subseteq D$ implies $X\mathbin{\triangle}Y\subseteq D$, which is clear because $$ X\mathbin{\triangle}Y\subseteq X\cup Y $$

A lower level proof. Let $x\in A\mathbin{\triangle}C$. Either “$x\in A$ and $x\notin C$” or “$x\in C$ and $x\notin A$”.

First case: $x\in A$ and $x\notin C$. If $x\in B$, then $x\in B\mathbin{\triangle}C$. If $x\notin B$, then $x\in A\mathbin{\triangle}B$.

Second case: similar.

Answer 2

This is actually a very easy proof. I think you are getting lost among all those fancy math symbols

Let $x \in A\triangle C$. Therefore $x \in A$ and $x \notin C$ are both true.

If $x \in B, x \in B \triangle C \subseteq D$

Otherwise, $x \notin B$. This implies $x \in A \triangle B \subseteq D$