no. A subset would have to allow for values in its parent which are not in its self.
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NO- by definition a set is not a proper subset of itself . ( It is a subset, but not a proper one. )
Let A be the set {1,2,3,4} B is {1,2} and B is a proper subset of A C is {1} and C is also a proper subset of A. B and C are proper subsets of the set A because they are strictly contained in A. necessarily excludes at least one member of A. The set A is NOT a proper subset of itself.
A is a subset of a set B if every element of A is also an element of B.
The line, itself, is a subset (though not a proper subset). A ray is a subset of a line with one end-point which extends in only one direction. A line segment is a subset of a line with two end points. A point is a subset of a line.
The set of Rational Numbers is a [proper] subset of Real Numbers.