Subset

In set theory, a subset is a set containing some or all members of another set.

For example, if the set S is defined as { a, b, c }, then { a }, { a, c } and { a, b, c } are all subsets of S.

The symbol ⊆ is used to indicate a subset. So, if S is a subset of T, then S ⊆ T. The symbol is sometimes read as “subset or equal to”, but in general, sets that are equal are subsets of each other.

The formal definition for a subset is:

S ⊆ T ↔ ∀x(x∈S → x∈T)

That is, S is a subset of T if and only if every element of S is also an element T.

Proper Subset

A proper subset is a subset which contains fewer elements of its parent set. For example, if S is defined as { a, b, c }, { a, b, c } and { a, b, c } are subsets of S, but { a, b, c } is not, since it is equal to S.

In other words, the term proper subset can be read as “subset of but not equal to ”.

The symbol ⊂ is used to indicate a proper subset. So, if S is a proper subset of T, then S ⊂ T.

The formal definition for a subset is:

S ⊆ T ↔ (∀x(x∈S → x∈T) ∧ S ≠ T)

That is, S is a subset of T if and only if every element of S is also an element T.

Set Equality

Set equality, the relation between two sets that have the same elements, can be defined by way of subsets:

S = T if and only if S ⊆ T and T ⊆ S.

The Empty Set

Another important note to remember is that the empty set, ∅, is a subset of every set, because the antecedent condition of the subset definition (x∈∅) is always false, thus making the conditional true for every x.