Question 835126: Prove that if a set is an improper subset of another set, then the two set are equal
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let X and Y be two sets. Let X be a subset of Y. So everything in X is also found in Y.
If X is a subset then the only two things are possible: it is a proper subset or it is an improper subset.
If it is a proper subset, then X will have less than Y (since X will be smaller).
If X is an improper subset, then X will have the same number of items in set Y. This will force X to be equal to Y.
If the two weren't equal for instance, then Y would have some element that is not in X, but that would make Y larger and hence X would be a proper subset.
However, X is an improper subset, which would again force Y to not have that extra element. So this proves that X = Y.
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