Question

If \(A,B,C,\)and \(D\)are sets, does it follow that \((A \oplus B) \oplus (C \oplus D) = (A \oplus C) \oplus (B \oplus D)\)?

Short answer

Yes, the given statement \((A \oplus B) \oplus (C \oplus D) = (A \oplus C) \oplus (B \oplus D)\) is true

Step-by-step solution

Step 1

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

Now,\((A \oplus B) \oplus (C \oplus D) = A \oplus (B \oplus (C \oplus D))\)

We know that \( \oplus \)is associative

\(\begin{aligned}{l} = A \oplus (B \oplus (D \oplus C))\\ = A \oplus ((B \oplus D) \oplus C)\end{aligned}\)

Since \( \oplus \)is commutative

\(\begin{aligned}{l} = A \oplus (C \oplus (B \oplus D))\\ = (A \oplus C) \oplus (B \oplus D)\end{aligned}\)

Hence, the given statement is true.