Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\Sigma$ be an infinite set.

Let $A,B \subseteq \Sigma$ be of finite symmetric difference iff they have a finite difference, more formally:

$A \sim B$ iff $|A \Delta B| \in \mathbb{N}$

How many equivalence classes does $\sim$ have?

share|improve this question
What you’re calling difference is generally called symmetric difference. –  Brian M. Scott Sep 18 '12 at 18:56
@Brian, thanks, I added "symmetric". –  Kaveh Sep 18 '12 at 19:07
add comment

1 Answer

up vote 3 down vote accepted

Let $A \subseteq \Sigma$ be an arbitrary set. Then the size of $[A]$ (the equivalence class of $A$) is equal to the number of finite subsets of $\Sigma$. The number of finite subsets of $\Sigma$ is of the same cardinality as $\Sigma$ since $\Sigma$ is an infinite set. Therefore we have $|[A]| = |\Sigma|$ for all $A$.

Let $I$ be the set of equivalence classes. We have $P(\Sigma) = \bigcup_{[A]\in I} [A]$ and distinct equivalence classes are disjoint. Therefore we have $2^{|\Sigma|} = |P(\Sigma)| = |\bigcup_{[A]\in I} [A]| = |I| |\Sigma|$. Thus by cardinal arithmetic we have $|I| = 2^{|\Sigma|}$.

share|improve this answer
Looks fine to me. –  Brian M. Scott Sep 18 '12 at 18:56
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.