0
$\begingroup$

For any three sets $A, B$ and $C$, show that $$A\Delta B = C \iff A = B\Delta C.$$

I am a student and wish some more information on the above. Kindly help.

  • 2
    Do you know that $\Delta$ is associative and $A\Delta A=\emptyset$?2012-08-28
  • 3
    What is $\Delta$ here? Symmetric difference?2012-08-28
  • 1
    @HenningMakholm: Yes, that is the customary meaning of the symbol.2012-08-28
  • 1
    Not so, @HaraldHanche-Olsen...according to WA (http://mathworld.wolfram.com/SymmetricDifference.html ), one must dispose of that symbol as it is already used in other contexts.2012-08-29
  • 2
    @DonAntonio: right. I will bow to the authority and will immediately stop following a custom that goes back to the beginnings of measure theory and set theory just because W|A (of all people) says so... FWIW: I've never seen any of the other symbols (except maybe +) that are suggested on that page in any measure theory or topology book. Deprecated. That's ridiculous.2012-08-29
  • 0
    As you wish. That symbol you say was introduced in the beginnings of set theory is said to be introduced less than two decades ago. I really don't care as long as it's made clear. You can read here http://www.proofwiki.org/wiki/Definition:Symmetric_Difference about other symbols. Besides this:(1) Jean Rubin's "set Theory for the mathematician" uses the symbol $\,\square\,$ , (2) Kaplansky's "Set theory and matrices" and Halmos's "Axiomatic Set theory" both use the symbol $\,+\,$ , (3) Suppes's " Axiomatic Set Theory" uses the symbol $\,\div\,$...so there's hardly "a custom".2012-08-29
  • 1
    @DonAntonio: Different cultures, I guess :) Measure theory is more my business than set theory and I thought Hausdorff used the symbol but I seem to have misremembered; sorry about that. I should probably have added the qualifier *modern* to measure theory and left set theory out. Virtually every measure theory book I know features that symbol and nothing else (some use + instead). Halmos's 1950 first edition of his *Measure theory* has it and it's in Royden's 1967 edition, just two examples among many others (and far older than 2 decades ago). Definitely a custom in measure theory.2012-08-29

5 Answers 5