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