Subset (⊆)
A set A is a subset of set B if all elements of A are also elements of B.
Notation: A ⊆ B
Example: If A = {1, 2} and B = {1, 2, 3, 4}, then A ⊆ B.
Proper Subset (⊂)
A set A is a proper subset of set B if all elements of A are in B, but B has at least one element not in A.
Notation: A ⊂ B (some texts use ⊆ with A ≠ B)
Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊂ B.
Improper Subset
An improper subset is either the set itself or the empty set.
Notation: When A ⊆ B and either A = B or A = ∅
Example: For set B = {1, 2, 3}, the improper subsets are:
∅ and {1, 2, 3}