|
Given two sets and , we say that is a subset of (which we denote as
or simply
) if every element of is also in . That is, the following implication holds:
The relation between and is then called set inclusion.
Some examples:
The set
is a subset of the set
because every element of is also in . That is,
.
On the other hand, if
, then neither
(because but
) nor
(because but
). The fact that is not a subset of is written as
. Similarly, we have
.
If
and
, it must be the case that .
Every set is a subset of itself, and the empty set is a subset of every other set. The set is called a proper subset of , if
and . In this case, we do not use
.
|