the difference between a subset and a proper subset
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A subset of a set S can be S itself. A proper subset cannot.
I don't think such a term is used in set theory. A proper subset, on the other hand, is a subset of the set, that is not equal to the set itself. The difference is comparable to the difference between "greater than" and "greater-or-equal", for real numbers.
S is a proper subset of T ifall elements of S are in T andthere is at least one element of T which is not in S.S is an improper subset if the second condition does not apply.
There is no difference between improper subset and equal sets. If A is an improper subset of B then A = B. For this reason, the term "improper subset" is rarely used.
Since ASCII ⊊ unicode, I don't know if there are ASCII codes for subset and proper subset. There are Unicode characters for subset and proper subset though: Subset: ⊂, ⊂, ⊂ Subset (or equal): ⊆, ⊆, ⊆ Proper subset: ⊊, ⊊,