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 (or is included in) Y;
- X ⊆ Y;
- Y is a superset of (or includes) 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. The relation "is a subset of" is called inclusion.
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.
BambooWeb 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: Every set A is a subset of itself. (reflexivity)
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. (antisymmetry)
PROPOSITION 3: 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. (transitivity)
PROPOSITION 4: 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.
Propositions 1, 2 and 3 show that ⊆ is a partial order on the class of all sets, and Propostion 4 shows that {} is the least element for this partial order.
