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)