Subset
We say a set is a subset of another set
if every element of
is also an element of
, and we denote this by $A \sub B$ (Error compiling LaTeX. Unknown error_msg). The empty set is a subset of every set, and every set is a subset of itself. The notation
emphasizes that
may be equal to
, while
says that
is any subset of
other than
itself.
The following is a true statement:
$\emptyset \sub \{1, 2\} \sub \mathbb{N} \sub \mathbb{Z} \sub \mathbb{Q} \sub \mathbb{R} \sub \mathbb{C} \sub \mathbb{C}\, \cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\}$ (Error compiling LaTeX. Unknown error_msg)
The set of all subsets of a given set is called the power set of
and is denoted
.