Subset : Proper subset 

If X and Y are sets and every element of X is also an element of Y, then we say or write:

• X is a subset of Y;
• X ⊆ Y;
• Y is a superset of X;
• Y ⊇ X.

Every set Y is a subset of itself. A subset of Y which isn't equal to Y is called proper. If X is a proper subset of Y, then we write X ⊂ Y. Analogous comments apply to supersets.

Notational variations

There are two major systems in use for the notation of subsets. The older system uses the symbol "⊂" to indicate any subset and uses "⊊" to indicate proper subsets. The newer system uses the symbol "⊆" to indicate any subsets and uses "⊂" to indicate proper subsets. Wikipedia uses the newer system, which can be handled by a wider variety of web browsers. Analogous comments apply to supersets.

Examples

• The set {1,2} is a proper subset of {1,2,3}.
• The set of natural numbers is a proper subset of the set of rational numbers.
• The set {x : x is a prime number greater than 2000} is a proper subset of {x : x is an odd number greater than 1000}
• Any set is a subset of itself, but not a proper subset.
• The empty set, written {}, is also a subset of any given set Y. (This statement is vacuously true.) The empty set is always a proper subset, except of itself.