# 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.

Jan 31, 2018

The proposition is true

#### Explanation:

1.- Let $x \in \left(A - B\right) \cup \left(C - B\right)$
That means $x \in \left(A - B\right) \mathmr{and} x \in \left(C - B\right)$
if $x \in \left(A - B\right)$ that means $x \in A$ and $x \notin B$
if $x \in \left(C - B\right)$ that means $x \in C$ and $x \notin B$
Thus $x \in \left(A \cup C\right) - B$
We have proven that $\left(A - B\right) \cup \left(C - B\right) \subset \left(A \cup C\right) - B$
2.- Let $x \in \left(A \cup C\right) - B$
That means $x \in A \cup C$ 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 $x \in \left(C - B\right)$
Thus we have $x \in \left(A - B\right) \cup \left(C - B\right)$
We have proven that $\left(A \cup C\right) - B \subset \left(A - B\right) \cup \left(C - B\right)$
Both inclusions are true, so
$\left(A - B\right) \cup \left(C - B\right) \subset \left(A \cup C\right) - B$
QED (Quod Erat Demonstrandum in Latin)