Question

Suppose that\(A,B,\)and\(C\)are sets such that\(A \oplus C{\bf{ = }}B \oplus C\). Must it be the case that\(A{\bf{ = }}B\).

Short answer

Yes, we conclude that \(A,B,\)and \(C\)are sets and the problem is symmetric in \(A\)and \(B\).

Step-by-step solution

Introduction

Given: \(A \oplus B = B \oplus C\)

Which is same as \(A \cup C - A \cap C = B \cup C - B \cap C\).

If \(A\)is empty, then it follows that \(B = \emptyset \)

Thus, we can assume that \(A\)is non-empty

Let \(x \in A.\)

Solve for the case \(x \in C\)

Consider \(x \in C\)

Then\(x \in A \cap C\)that implies that\(x \notin A \oplus C\).

Which implies from the above equality that\(x \in B \oplus C\).

However, \(x \in C\)

Thus, the only way \(x \notin B \oplus C\)is if \(x \in B \cap C\)

i.e., \(x \in B\).

Solve for the case \(x \notin C\)

Consider \(x \notin C\)

Then\(x \notin A \oplus C\)which implies that the first equality that\(x \in B \oplus C\)

But \(x \notin C\)

Thus, the only way \(x \in B \oplus C\)is if \(x \in B\)

Final Answer

Thus, we have proved that \(x \in A\)then \(x \in B\). We conclude that the problem is symmetric in \(A\)and \(B\).