Question

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.

Answer 1

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)