Is x∈ ((A\C) ∩ B) ∪ ((A\B) ∩ C) same as x∈ (AΔC)\B ?
I need to prove that $(A\backslash (B \cup C))\cup (C\backslash (A\cup B))$ $=$ $(A\Delta C)\backslash B $ and using 'Let x∈... ...' method I get to the point where x∈ ((A\C) ∩ B) ∪ ((A\B) ∩ C), so is that same as x∈ (AΔC)\B?
Can anyone help?
Thank you!
You can also use the properties of unions, intersections and difference of two sets to prove that, without involving elements of the set and much calculation.
LHS = $ (A\(B\cup C))\cup (C\(A\cup B))$
=$(A\cap(B\cup C)^c)\cup(C\cap (A\cup B)^c)$
=$(A\cap B^c\cap C^c)\cup (C\cap A^c \cap B^c)$
=$((A\cap C^c)\cup (A^c \cap C))\cap B^c$
=$(A\Delta C)\B$
=RHS
Hope it helps:)