Proper 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.