For my assignment, one of the questions is 'how many equivalence classes are there for this relation and describe them.' I have come to the conclusion that there are an infinite number of equivalence classes, but I would like to know whether this is possible, and therefore correct. I don't want to copy and paste the question here -- lest I be accused of cheating -- so I will rephrase the question:
The set T has a cardinality of infinity, and contains elements (I will call them horses) that are related to one another iff they contain the same number of hairs. Note: The number of hairs on a horse may be infinite.
So horses are related iff they have the same number of hairs. That means that two horses are related to one another iff they both contain the same number of hairs out of the set of 0-infinity number of hairs, thus the equivalence classes are 0,1,2,3,4,...,infinity.
Thanks for reading.