Chapter 2 Set Theory

Proper Subset

A Proper Subset is when set A is a subset of set B but they are not equal sets. {1,3} ⊂ {1,3,5} In some examples both the subset and proper subset symbols can be used. This is one such example.

set

A collection of objects whose contents can be clearly determined

Infinite Set

A set that goes on and on. . .

Finite Set

A set with a definite number of elements Examples: Z = { 1, 2, 3, 4} n(a) n(a) is a natural number

Set A is not a subset of B

A ⊄ B

Set A is a subset of set B

A ⊆ B (of every element of set A is also an element of set B)

Equal Set

Equal Sets have exactly the same elements, regardless of order or possible repitition of elements. If two sets are equal they must also be equivalent. Example: {O, L, D} = {L, D, O}

Equivalent Sets

Equivalent sets have the same number of elements, or the same cardinality. A one-to-one correspondence between sets A and B means that each element in A can be paired with exactly one element in B, and vice versa.

∩

Intersection Objects that belong to set A and set B

And and But

Mean intersection

Not

Means compliment

Or

Means union

Subset

Set A is a subset of set B, expressed as A ⊆ B If every element in set A is also in set B.

Chapter 2

Set Theory

Tattoo Subset Example

Subset A = {x | x is a tattoed American between the ages of 25-29} Set B = {x | x is a tattoed American}

Number of Proper Subsets

The number of proper subsets of a set with n elements is 2^n − 1

Number of Subsets

The number of subsets of a set with n elements is 2^n

elements/members

The objects in a set

∪

Union Members of set A and set B, or both

Set-Builder Notation

W = { x | x is a day of the week}

Roster Method

W = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

Word Description

W is the set of the days of the week

Cardinal Number of the Union of Two Finite Sets

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Number of Distinct Elements in a Set

n(A)

The null element inside a set

{Ø}

The Empty Set or Null Set

Ø The empty set is a subset of every set.

The Set of Natural Numbers

ℕ = { 1, 2, 3, 4, 5. . .} (Also known as the counting numbers)

Is an element of

∈

Is not an element of

∉