Subset
A set X is a subset of a set Y if all elements of X are also in Y.
Examples:
- 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. Also, 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 it the empty set is also an element of X, for any given set X.
If the above paragraph sounds a bit like double-speak - it is only because it is stating the obvious.
See also: