1.3.4.1 Proper Subset and Superset

If all the elements of  are also elements of , but  contains at least one additional element that is not in , then  is a proper subset of , denoted by  ⊂ . Equivalently, we can also say that  is a proper superset of , denoted by  ⊃ .

Example 1.11. To illustrate this, consider a set  = {1, 2, 3}, then its proper subsets are:

• {1}
• {2}
• {3}
• {1, 2}
• {1, 3}
• {2, 3}
• { }

{ } is an empty set, covered in detail in section 1.3.6.

Equivalently,  is a proper superset of the aforementioned sets.