How do you prove this? 7) For sets A,B,C prove (A - B) ∪ (C - B) = (A ∪ C) - B by showing Left side ⊆ Right side and Right side ⊆ Left side.

1 Answer
Jan 31, 2018

The proposition is true

Explanation:

1.- Let #x in (A-B)uu(C-B)#
That means #x in (A-B) or x in (C-B)#
if #x in (A-B)# that means #x in A# and #x notin B#
if #x in (C-B)# that means #x in C# and #x notin B#
Thus #x in (A uu C)-B#
We have proven that #(A-B)uu(C-B) sub (AuuC)-B#
2.- Let #x in (AuuC)-B#
That means #x in AuuC# but #x notin B#
So #x in A# or #x in C# but in both cases #x notin B#
That means #x in A-B# or #x in (C-B)#
Thus we have #x in (A-B)uu(C-B)#
We have proven that #(AuuC)-B sub (A-B)uu(C-B)#
Both inclusions are true, so
#(A-B)uu(C-B) sub (AuuC)-B#
QED (Quod Erat Demonstrandum in Latin)