Null Universal Proper and Subset
In this class, We discuss Null Universal Proper and Subset.
The reader should have prior knowledge of the set theory introduction. Click Here.
NULL Set:
A set that does not contain any of the elements is called a null set.
We represent the null set using two ways
A = { }
A = Φ
A = {x/x is an integer and 2<x<3}
Universal Set: We consider a universal set if it contains every set under discussion.
We represent the universal set using U or E.
Subset or Set inclusion:
X and Y are two sets
If every element of X is an element of Y, then X is a subset of Y.
X ⊆ Y
X = {a, b, c}
Y = {a, b, c, d}
X ⊆ Y
Properties of set inclusion:
1) Every set is a subset of itself
A ⊆ A reflexive
2) An empty set is a subset of every set.
3) Set inclusion is transitive
Proper subset / proper inclusion
X and Y are two sets
X proper subset Y if X ⊆ Y and at least one element extra is needed in Y.
X = {a, b, c}
Y = {a, b, c, d}
X ⊂ Y
Properties of Proper Subset:
1) Proper subset is not reflexive
2) Proper subset is transitive