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Proposition: Two sets have same size, if there is a bijection between:
A and B are of same size : <=> exists f: f: A ->bijective B.One set is not larger than another set, if there exists an injection from the first set into the second set:
A is not larger than B : <=> exists f: f: A ->injective B.One set is smaller than another set, if they are not of same size and the second one is not larger than the first one.
A is smaller than B : <=> (A is not larger than B) /\ ~(A and B have same size).