Using the choose-an-element method, prove that A - (B ∪ C) ⊆ (A ∪ B) - C.
To prove this, we will choose an arbitrary element x from the set A - (B ∪ C) and show that it is also an element of (A ∪ B) - C.
Let x be an arbitrary element in A - (B ∪ C). This means that x is in set A but not in the union of sets B and C. In other words, x is in A but not in either B or C.
Now, we want to show that x is also an element of (A ∪ B) - C. This means that x is in the union of sets A and B, but not in set C.
Since x is in A, it is also in the union of sets A and B. Therefore, x is in (A ∪ B).
Now, we need to show that x is not in set C. Since x is not in either B or C, it is not in set C.
Therefore, we have shown that x is in (A ∪ B) - C.
Since x was an arbitrary element chosen from A - (B ∪ C), we can conclude that A - (B ∪ C) ⊆ (A ∪ B) - C.
This completes the proof.