Let \(A, B, C \subset X\) and denote by \(A \triangle B\) the symmetric difference as in Problem 2.2 . Show that

(i)

(ii) \(A \Delta(B \triangle C)=(A \triangle B) \triangle C\);

(iii) \(\mathscr{P}(X)\) is a commutative ring (in the usual algebraists' sense) with 'addition' \(\triangle\) and 'multiplication' \(\cap\).

[ use indicator functions for (ii) and (iii).]

Data from problem 2.2

 Let \(A, B, C \subset X\). The symmetric difference of \(A\) and \(B\) is \(A \triangle B:=(A \backslash B) \cup(B \backslash A)\). Verify that

\[(A \cup B \cup C) \backslash(A \cap B \cap C)=(A \triangle B) \cup(B \triangle C)\]

Transcribed Image Text:

1AAB=1A +1B-21A1B=114-1B| = 14 +18 mod 2;