# Set Theory

## Subsets

### Terms

- subset,
*n.*
- (
*math*) The set A is a subset of set B iff every member of A is also a member of B.
- proper subset,
*n.*
- (
*math*) 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.
- proper superset,
*n.*
- (
*math*) 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,
*n.*
- (
*math*) Two sets A and B are equal
iff A is a subset of B,
and B is a subset of A.
- superset,
*n.*
- (
*math*) The set A is a superset of set B
iff every member of B is also a member of A.

### Symbols

- ⊂
- is a proper subset of
- ⊆
- is a subset of
- ⊃
- is a proper superset of
- ⊇
- is a superset of
- ⊄
- is not a subset of

### Notation

- The symbol ⊆ (or ⊇, reversed) is used do 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 do 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 do denote "not a subset":

if `A = {r, s, t}`,

and `B = {r, s, u, v}`, and

`A ⊄ B`, and