Symmetric difference: Need help to prove the complement 
I need to prove that $$A\Delta B = A^C\Delta B^C$$ by using an element to prove the right side of the sentence is equal to the left side. 
  For example : $$(X+Y) = (X+Y)$$
  i know how to prove that without using elements:
  $$A\Delta B = A^C\Delta B^C$$ $$(A-B) \cup (B-A) = (A-B)^C \cup (B-A)^C$$
  $$(A\land B^C) \lor (B\land A^C) = (A\land B^C)^C \lor (B \land A^C)^C$$
  $$(A\land B^C) \lor (B\land A^C) = (A^C\land B) \lor (B^C \land A)$$
  and by changing their places ( which i can do) i  can get that proof:
  $$(A\land B^C) \lor (B\land A^C) = (A\land B^C) \lor (B\land A^C)$$
  But for now i need to prove that by using elements and show that the elements are equal both sides, thanks for helping me :)

 A: With elements, we can just appeal to a more "intuitive" definition of $A\triangle B$. An element $x$ is in  $A\triangle B$ if and only if $x$ is in either $A$ or $B$, but not both. On the other hand, $x$ is in $A^c\triangle B^c$ if and only if $x$ is in $A^c$ or $B^c$, but not both. Any $x$ is in $A^c$ but not $B^c$ if and only if $x$ is in $B$, but is not in $A$. For the other possibility, $x$ is in $B^c$ but not in $A^c$ if and only if $x$ is in $A$, but not in $B$. Therefore an element $x$ is in $A^c\triangle B^c$ if and only if it is in one of $A$ or $B$, but not both. We have arrived at the same definition by elements for the two sets.
A: For any $x$:
$$x \in A \Delta B \text{ if and only if}$$
$$\text{either } x \in A \text{ and } x \not \in B \text{ or } x \not \in A \text{ and } x \in B \text{ if and only if}$$
$$\text{either } x \not \in A^C \text{ and } x \in B^C \text{ or } x \in A^C \text{ and } x \not \in B^C \text{ if and only if}$$
$$\text{either } x \in A^C \text{ and } x \not \in B^C \text{ or } x \not \in A^C \text{ and } x \in B^C \text{ if and only if}$$
$$x \in A^C \Delta B^C$$
A: All you needed was: $$A\Delta B ~{= (A\cap B^\complement)\cup(B\cap A^\complement)\\= (B^\complement\cap (A^\complement)^\complement)\cup(A^\complement\cap (B^\complement)^\complement) \\ = A^\complement\Delta B^\complement}$$
Then put this into words:
By definition of symmetric difference: any element of $A\Delta B$ is either in $A$ but not $B$, xor it is in $B$ but not $A$.   By definition of complements, that is the same as claiming it is in $B^\complement$ but not $A^\complement$ xor it is in $A^\complement$ but not $B^\complement$.   Therefore if an element is in $A\Delta B$, then it is in $A^\complement\Delta B^\complement$ and vice versa by symmetry.
That is all.
