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