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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 is not 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.
Simple results
PROPOSITION 1: Given any three sets A, B and C, if A is a subset
of B and B is a subset of C, then A is a subset of C.
PROPOSITION 2: Two sets A and B are equal if and only if A is a subset of
B and B is a subset of A.
PROPOSITION 3: The empty set is a subset of every set.
Proof: Given any set A, we wish to prove that {} is a subset of A. This involves showing that all
elements of {} are elements of A. But there are no elements of {}.
For the experienced mathematician, the inference "{} has no elements, so all elements of {} are elements of A" is
immediate, but it may be more troublesome for the beginner. Since {} has
no members at all, how can "they" be members of anything else? It may help to think of it the other way around. In order to prove
that {} was not a subset of A, we would have to find an element of {} which was not also an element of
A. Since there are no elements of {}, this is impossible and hence {} is indeed a subset of A.
These propositions show that ⊆ is a partial order on the class
of all sets, and {} is a bottom element.
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