Subsets

subset

The set A is a subset of set B iff every member of A is also a member of B.

proper subset

The set A is a proper subset of set B iff A is a subset of B, but B is not a subset of A.

superset

The set A is a superset of set B iff every member of B is also a member of A.

proper superset

The set A is a proper superset of set B iff A is a superset of B, but B is not a superset of A.

set equality

Two sets A and B are equal iff A is a subset of B, and B is a subset of A.

Axiom of Empty Set

There exists a set with no elements

The empty set is a subset of all sets.

⊂

is a proper subset of

⊆

is a subset of

⊃

is a proper superset of

⊇

is a superset of

⊄

is not a subset of

The symbol ⊆ (or ⊇, reversed) is used to denote the subset/superset relationship:
if A = {r, s, t},
and B = {r, s, t, u, v}, and
A ⊆ B, and
B ⊇ A

The symbol ⊂ (or ⊃, reversed) is used to denote a proper subset/superset relationship:
if A = {r, s, t},
and B = {r, s, t, u, v}, and
A ⊂ B, and
B ⊃ A

The symbol ⊄ is used to denote "not a subset":
if A = {r, s, t},
and B = {r, s, u, v}, and
A ⊄ B, and