Set Convergence and Subsets

subset

occurs when all the elements of a set A are in the set B

Subset of all numbers

N ⊆ Z ⊆ Q ⊆ R

proper subset

occurs when all the elements of set A are in the set B, BUT A ≠ B

Differentiate ⊆ and ⊂

• ⊂ represents a proper set

• ⊆ represents a set

power set

set of all subsets

What is P(A) given A = {1, 2, 3}

P(A) = { Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

The cardinality of any power set A of n elements is ___

2ⁿ

intersection of sets

elements that are in both sets

{ x ∈ U | x ∈ A and x ∈ B }

What is the intersection of the following sets?

A = [1, 10)

B = (4, 16)

A ∩ B = (4, 10)

What is the intersection of the following set?

A = { x ∈ Z; x + 2 >= 3 }

B = R

A ∩ B = A

[A ∩ B = A <=> A ⊆ B]

union of sets

element that are in either set

{x ∈ U | x ∈ A or x ∈ B}

What is the union of the following set?

A = {a, b, c, d}

B = {a, b, e, f, g}

A ∪ B = {a, b, c, d, e, f, g}

[Elements cannot repeat]

What is the union of the following set?

A = [3, 17)

B = (0, 20)

A ∪ B = (0, 20) = B

[ A ∪ B = B <=> A ⊆ B ]

For any set A and the universal set U:

A ∪ U =

U

A ∩ U =

A

A ∩ Ā =

Ø

A ∪ Ā =

U

Set disjoint

Two sets have no common elements

A ∩ B = Ø