# Symmetric Difference

• Sep 3rd 2009, 02:09 AM
ninni
Symmetric Difference
The symmetric difference of A and B is
(a) \$\displaystyle (A-B) \cap (B-A)\$
(b) \$\displaystyle (A-B)\cup (B-A)\$
(c) \$\displaystyle (A \cup B) - (A \cap B)\$
(d) \$\displaystyle \{(A \cup B)-A\} \cup \{(A \cup B)-B\}\$

Its answer given in the book is (b). But I found that (c) and (d) are also true. Am I right? If not please tell why. Thanks a lot for giving me your time.
• Sep 3rd 2009, 04:49 AM
Symmetric Difference
Hello ninni
Quote:

Originally Posted by ninni
The symmetric difference of A and B is
(a) \$\displaystyle (A-B) \cap (B-A)\$
(b) \$\displaystyle (A-B)\cup (B-A)\$
(c) \$\displaystyle (A \cup B) - (A \cap B)\$
(d) \$\displaystyle \{(A \cup B)-A\} \cup \{(A \cup B)-B\}\$

Its answer given in the book is (b). But I found that (c) and (d) are also true. Am I right? If not please tell why. Thanks a lot for giving me your time.

You are right. (a) is the only expression not equal to the symmetric difference of the sets A and B.

• Sep 5th 2009, 01:09 PM
Researcher
Quote:

Originally Posted by ninni
The symmetric difference of A and B is
(a) \$\displaystyle (A-B) \cap (B-A)\$
(b) \$\displaystyle (A-B)\cup (B-A)\$
(c) \$\displaystyle (A \cup B) - (A \cap B)\$
(d) \$\displaystyle \{(A \cup B)-A\} \cup \{(A \cup B)-B\}\$

Its answer given in the book is (b). But I found that (c) and (d) are also true. Am I right? If not please tell why. Thanks a lot for giving me your time.

Yes you are right about (c).[I dindn'ttry to prove the others].
One of the proofs of (c):
http://www.picamatic.com/show/2009/0...0_bigthumb.JPG