For each relation $p$ described below, determine if $p$ is reflexive, symmetric, transitive, anti-symmetric. In each case, if $p$ is an equivalence relation, describe the equivalence classes.

a) $A = P(\Bbb{Z})$ (the power set of integers). Let $X$ be a subset of $\Bbb Z$. Define $p$ on $A$ by $BpC$ if and only if $B\cap X = C \cap X$

b) Two sequences of real numbers $(a_n)$ and $(b_n)$ are eventually equal if there exists some $K$ element of integers such that $a_k = b_k$ for all $k \ge K$. Let $A$ be the set of all sequences of real numbers, and define $p$ by $(a_n)p(b_n)$ if and only if $(a_n)$ and $(b_n)$ are eventually equal.

I have determined that a) is an equivalence relation as it has the transitive, reflexive properties. Although not sure what to write as for the equivalence class

and b) confuses me to all hell.

any help will be greatly appreciated.

I'm not sure how to do these, I am very tired so clear instructions would be great! Thanks guys.

  • $\begingroup$ For b), where do you get confused? Do you understand what "eventually equal" means? $\endgroup$ – rogerl Oct 22 '14 at 1:33
  • $\begingroup$ No not really.. but im guessing that if i were to have to find out if it were symmetrical it would go something like this $(an)P(bn)$ if and only if $(an)=(bn)$ therefore $(bn) = (an)$ therefore $(bn)P(an)$ thus, it is symmetrical. $\endgroup$ – I hate math Oct 22 '14 at 1:39
  • $\begingroup$ Well, sure, but that's just the definition of what a symmetric relation is. You haven't really used the definition of the relation at all. $\endgroup$ – rogerl Oct 22 '14 at 1:42
  • $\begingroup$ But if i can determine that P is symmetrical is that not answering the questions? Or atleast part of. $\endgroup$ – I hate math Oct 22 '14 at 1:45
  • $\begingroup$ I feel like this question was posted before, then got deleted and re-posted. $\endgroup$ – Asaf Karagila Oct 22 '14 at 1:55

Hints: a) $ApB$ means that $A\cap X = B\cap X$. So any two sets that are equivalent have the same intersection with $X$.

b) Eventually equal means that if you were to write out the sequences, one on top of the other, that once you get far enough out, the two sequences are identical. So is it clear that $(a_n)p(a_n)$? Is it clear that it's symmetric? Transitivity is a touch harder, but try writing out three such sequences $(a_n)p(b_n)$ and $(b_n)p(c_n)$ and see if you can figure out why $a_n$ and $c_n$ agree after some point.


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