$x,y$ related if and only if $x\cap\{{1,3,5\}}=y\cap\{{1,3,5\}}$. Let A be the power set of $\{1,2,3,4,5\}$, let $z= \{1,2,3\}$, and let $(x,y) \in R$ if and only if
$$x \cap \{1,3,5\} =  y \cap \{1,3,5\}$$
I'm supposed to find the equivalence class, number of equivalence class and determine which equivalence class the element $z$ belongs to.
This is not homework. It's a practice textbook question, and I have no idea how to even start on this problem. Mainly, I don't understand the relation.
 A: X,Y are elements of the power set A. 
Please read the definition of power set, 
the equivalent relation should be X~X $\cup${2}~X$\cup${4}~X $\cup${2,4}.
The number of equivalent class equal to the cardinality of the power set of {1,3,5}, that is 8.
Each equivalent class contain 4 members. 
for example: {1,5}, {1,2,5}, {1,4,5},{1,2,4,5} form a equivalent class.
A: Just to help a bit. The set $A$, is the set of all subsets of $\Omega = \{1,2,3,4,5\}$. So for example as an element you would have $\{1,2\} \in A$ or the empty set $\emptyset\in A$.
Now you say that two such elements $X,Y\in A$ are equivalent (maybe written $X\sim Y$) if (and only if) $X\cap \{1,3,5\} = Y\cap \{1,3,5\}$. 
So you would for example have that $\{1,2,3\} \sim \{1,3,4\}$ because 
$$
\{1,2,3\}\cap \{1,3,5\} = \{1,3\} \\
\{1,3,4\}\cap \{1,3,5\} = \{1,3\}.
$$
That would mean that the elements $\{1,2,3\}\in A$ and $\{1,3,4\}$ are in the same equivalence class.
So one approach to this problem that might be helpful for you would be to


*

*Write down all the elements of $A$

*Write down what the intersection of each element in $A$ with $\{1,3,5\}$ is.

*The elements that have the same intersection are in the same equivalence class. 

*Use this list should answer all your questions.


(There are "rules" that could help you do this, but I think that the best thing would be to just write it all down. Maybe you see some things about the number of equivalence classes and such...)
