Subset
is a subset of , written , if every element of is an element of ; it is a proper subset when moreover .
Definition 2.3 (Set Inclusion, Equality, Subset and Proper Subset)
We write if is an element of , and if not.
Two sets are equal if they have the same elements. i.e. if and only if .
A set is a subset of , written or , if every element of is an element of .
is said to be a proper subset of if and . This is also written as .
Equality from inclusion
Two sets are equal exactly when each is a subset of the other: if and only if and . More generally, denotes the subset of comprising exactly those elements for which the property holds.
Related
Stated in
- Definition 2.3 (Set Inclusion, Equality, Subset and Proper Subset)§2.1.1 Introduction on Sets