Using property of sets prove that (A union B) minus A is equal to B- A?

Question

Answer

Proof:

Property of Sets:
The property we will use to prove the given statement is:
$ (A \cup B) - A = B - A $
Definition of Union:
$ A \cup B $ represents the set of all elements that are in either A or B or in both.
Definition of Minus:
$ X - Y $ represents the set of all elements that are in X but not in Y.
Proof:
To prove $ (A \cup B) - A = B - A $, we need to show that both sets have the same elements.
First Inclusion:
Let's first show that $ (A \cup B) - A \subseteq B - A $.
Take an element $ x $ from $ (A \cup B) - A $. This means $ x \in A \cup B $ and $ x \notin A $.
Since $ x \in A \cup B $, it implies that $ x \in B $ (as $ x \notin A $).
So, $ x \in B $ and $ x \notin A $, which implies $ x \in B - A $.
Therefore, $ (A \cup B) - A \subseteq B - A $.
Second Inclusion:
Now, let's show that $ B - A \subseteq (A \cup B) - A $.
Take an element $ y $ from $ B - A $. This means $ y \in B $ and $ y \notin A $.
Since $ y \in B $, it implies that $ y \in A \cup B $ (as $ y \notin A $).
So, $ y \in A \cup B $ and $ y \notin A $, which implies $ y \in (A \cup B) - A $.
Therefore, $ B - A \subseteq (A \cup B) - A $.
Conclusion:
Combining both inclusions, we have shown that $ (A \cup B) - A = B - A $.