Types of Sets
Read this page to become familiar with the various types of sets. This page is an aid to set terminology and notation.

Cardinality of Set X:

n(X), Set X contains n(X) elements

Empty:

holds no elements

{ }, Ø, n(X) = 0, A = 0

Equal:

two sets that contain the same elements, and no other elements

=, X = Y, Set X is equal to Set Y

Equivalent:

two sets not necessarily having the same elements, while having the same number of elements (the same cardinality)

n(X) = n(Y), Set X is equivalent to Set Y; ≈ , X ≈ Y

Finite:

holds a number of elements that is limited and countable

Improper Subset:

a set that is exactly the same as itself

Infinite:

holds a number of elements that is unlimited and uncountable

∞, n(X) = ∞

Null:

also known as the empty set – the set that holds no elements

{ }, Ø, n(X) = 0, A = 0

Proper Subset:

a set X that contains only elements of set Y but does not contain at least one element of Y

⊂, X ⊂ Y, Set X is a proper subset of Set Y

Proper Superset:

set X holds all elements of set Y but is not equal to Y

⊃, X ⊃ Y, Set X is a proper superset of Set Y

Power:

holds all subsets of a given set

P, X = P(Y), Set X contains all the subsets of Set Y

Singleton:

holds only one element, no more and no less

Subset:

a set X that contains only elements of set Y

⊆, X ⊆ Y, Set X is a subset of Set Y

Superset:

set X contains all elements of set Y, and only elements of Y

⊇, X ⊇ Y, Set X is a superset of Set Y

Universal:

holds all elements of all other sets under consideration

U, X = U, Set X is the universal set
Source: Saylor Academy
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