1.3.4.3 Subset Properties

Subsets have a few important properties:

• Every set is unconditionally the improper subset of itself, denoted by  ⊆  or  ⊆ .

Example 1.13. Say, If  = {1}, then {1} ⊆ {1}

        

• Empty set is a proper subset of every set.

Example 1.14. Say, If  = {1, 2}, then { } ⊂ {1, 2}

• Proper subset is transitive, which means if  is a proper subset of , and  is a proper subset of , then  is a proper subset of . So, if  ⊂  and  ⊂  then  ⊂  

Example 1.15. If  = {1, 2},  = {1, 2, 3},  = {1, 2, 3, 4}, then  ⊂