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A set X is a subset of a set Y if all elements of X are also in Y.


  • The set {1, 2} is a subset of {1, 2, 3}.
  • The set {1, 2} is also a subset of itself.
  • The set of natural numbers is subset of the rational numbers.
  • The set {x : x is a prime number greater than 2000} is a subset of {x : x is an odd number greater than 1000}

In case you were wondering, for any given set X, X is always considered to be a subset of itself (by definition) - a proper subset is any subset except the set itself. The empty set, written {}, is also a subset of any given set X. This is because the empty set vacuously satisfies the definition of a subset of X: since the empty set has no elements, every element in the empty set is also an element of X.

If the above paragraph sounds a bit like double-speak - it is only because it is stating the obvious.

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