Let U be a set. show that is a unique s that is an element of the power set of U so that S△A=U for all A that is an element of the power set of U
I wrote a logical statement of this but I am not sure how to proceed
To prove that there exists a unique set S in the power set of U such that S △ A = U for all A in the power set of U, you can use a proof by contradiction.
1. Assume that there are two sets, S1 and S2, in the power set of U such that S1 △ A = U and S2 △ A = U for all A in the power set of U.
2. Let's consider the set B = S1 △ S2. By the definition of symmetric difference, every element that is in S1 but not in S2 is in B, and every element that is in S2 but not in S1 is also in B.
3. Since S1 △ A = U for all A in the power set of U, it means that every element of U is either in S1 or in A, but not both. Similarly, S2 △ A = U implies that every element of U is either in S2 or in A, but not both.
4. Combining these two conditions, every element of U is either in S1 or in S2, but not both. This means that every element of U is in B (S1 △ S2) as well.
5. However, according to the definition of symmetric difference, if an element is in B, it must either be in S1 or in S2, but not both. This contradicts our previous conclusion that every element of U is in B.
6. Therefore, our assumption that there are two sets, S1 and S2, in the power set of U satisfying S1 △ A = U and S2 △ A = U for all A in the power set of U is false.
7. Hence, there must be a unique set S in the power set of U such that S △ A = U for all A in the power set of U.
Therefore, we have proven that there exists a unique set S in the power set of U satisfying S △ A = U for all A in the power set of U.