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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;
  • XY;
  • Y is a superset of X;
  • YX.

Every set Y is a subset of itself. A subset of Y which is not equal to Y is called proper. If X is a proper subset of Y, then we write XY. 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.



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