TITLE: What exactly are equivalence classes
QUESTION [6 upvotes]: What exactly are equivalence classes?  Suppose I have an equivalence relation $\sim$ on some set $X$ we denote this as $x \sim y$. The equivalence classes are then $[x] = \{y \in X : y \sim x\}$.  
However I have struggled as to what this actually mean, does this mean that for some $y \in X$ the equivalence classes are just the objects that satisfy the symmetric property?

REPLY [2 votes]: Think on this example:
You classifie your clothing in a wardrobe. 
You have different types of clothes (T-shirt, shoes, trousers...). If each kind of them are relationed with others clothings of the same type.
If you classifie each different type in a different part of the wardrobe (the parts of the wardrobe are the equivalence classes)