Final answer:
To prove A∆B = (A − B) ∪ (B − A), each element must be in either set A or B but not both, which corresponds to being in either (A − B) or (B − A), hence the two expressions represent the same set.
Step-by-step explanation:
The statement ∆ represents the symmetric difference between two sets A and B. To prove that A∆B = (A − B) ∪ (B − A), let's consider each element and which set it belongs to within the symmetric difference and the union of differences.
- An element in A∆B is in either A or B, but not in both.
- An element in (A − B) is in A but not in B.
- An element in (B − A) is in B but not in A.
- So, an element in (A − B) ∪ (B − A) is in A but not in B, or in B but not in A, which is exactly the condition for being in A∆B.
Therefore, A∆B consists of elements exclusive to A and to B, and thus it is the same as the union of the sets (A − B) and (B − A).