Question

prove it logically that symmetric difference is associative in sets​

Answer 1

Answer:

The symmetric difference AA △ BB of AA and BB is defined as (A−B)∪(B−A)(A−B)∪(B−A). (The set of elements that belong to A or B but not both.)

(a) Construct a truth table to show that △ is associative.

(b) Show that xx belongs to AA △ (B(B △ C)C) if and only if xx belongs to an odd number of the sets AA, BB and CC and use this observation to give a second proof that △ is associative.

For part (a), I'd be grateful is someone could post the full truth table, as I'm not entirely sure what column headings they are looking for.

For part (b), this is what I did:

(For the left to right implication) Let x∈x∈ AA △ (B(B△ C)C). Then, by the definition of △, we have either x∈x∈ AA or x∈x∈ (B(B △ C)C) but not both. So if x∈Ax∈Athen either it is not in BB nor