I tried to prove this equation $(A\bigtriangleup B)\cup C=(A\cup C)\bigtriangleup(B\setminus C)$ by elementhood and set algebra but with no result. I can see that equality stands in Venn's diagrams, and I also proved it with truth tables, but I would like to have solution with set algebra or elementhood. I would appreciate any pointers in solving this.

This is from Velleman's How to Prove It, chapter 1 section 4 exercise 13.

**Solution with set algebra**

After some time pounding this exercise, I came up with following solution:

$$(A\bigtriangleup B)\cup C=(A\cup C)\bigtriangleup(B\setminus C)\\
=((A\cup C)\cap (B\cap C^C)^C)\cup((B\cap C^C)\cap(A\cup C)^C)\\
=((A\cup C)\cap (B^C\cup C))\cup(B\cap C^C\cap A^C)\\
=(C\cup(A\cap B^C))\cup(B\cap C^C\cap A^C)\\
=C\cup(A\cap B^C)\cup(B\cap A^C)\\
=(A\bigtriangleup B)\cup C$$