Set theory

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Basic Definition[edit]

The term "set" can be thought as a well-defined collection of objects. In set theory, These objects are often called "elements".

  • We usually use capital letters for the sets, and lowercase letters for the elements.
  • If an element belongs to a set , we can say that " is a member of the set ", or that " is in ", or simply write .
  • Similarly, if is not in , we would write .

(Example) . In this case, is the element and is the set of all real numbers.

Notation[edit]

Example of a common notation style for the definition of a set:

  • is a set
  • is a property (The elements of may or may not satisfy this property)
  • Set can be defined by writing...

This would read as "the set of all in ,such that of ."

Elements[edit]

There are two ways that we could show which elements are members of a set: by listing all the elements, or by specifying a rule which leaves no room for misinterpretation. In both ways we will use curly braces to enclose the elements of a set. Say we have a set that contains all the positive integers that are smaller than ten. In this case we would write . We could also use a rule to show the elements of this set, as in .

In a set, the order of the elements is irrelevant, as is the possibility of duplicate elements. For example, we write to denote a set containing the numbers 1, 2 and 3. Or,.

Subsets[edit]

A is a subset of B.

Formal universal conditional statement: "set A is a subset of a set B"

  • , if , then

Negation:

  • such that and
If and only if: 
    for all :
        If: ( is an element of A)
        then: ( is an element of B)
then: 
    set A is a subset of set B

Truth Table Example:

if , then and
1 1 1 0
1 0 0 1
0 1 1 0
0 0 1 0

A proper subset of a set is a subset that is not equal to its containing set. Thus

A is a proper subset of B

Set Identities[edit]

Let all sets referred to below be subsets of a universal set U.

(a) A ∪ ∅ = A and (b) A ∩ U = A.

5. Complement Laws:

(a) A ∪ A c = U and (b) A ∩ A c = ∅.

6. Double Complement Law:

(A c ) c = A.

7. Idempotent Laws:

(a) A ∪ A = A and (b) A ∩ A = A.

8. Universal Bound Laws:

(a) A ∪ U = U and

(b) A ∩ ∅ = ∅.

Identity For all sets A
Identity Laws:

Complement Laws:


Double Complement Law:
Idempotent Laws:

Universal Bound Laws:
Identity For all sets A and B
Commutative Laws:

2. Associative Laws: For all sets A, B, and C,

(a) (A ∪ B) ∪ C = A ∪ (B ∪ C) and

(b) (A ∩ B) ∩ C = A ∩ (B ∩ C).

3. Distributive Laws: For all sets, A, B, and C,

(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and

(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

For all sets A and B,

De Morgan’s Laws:

(a) (A ∪ B) c = A c ∩ B c and (b) (A ∩ B) c = A c ∪ B c .

Absorption Laws:

(a) A ∪ (A ∩ B) = A and (b) A ∩ (A ∪ B) = A.

Set Difference Law:

  • A − B = A ∩ B c .

Complements of U and ∅:

Cardinality[edit]

The cardinality of a set is the number of elements in the set. The cardinality of a set is denoted

Types of Sets by Cardinality[edit]

A set can be classified as finite, countable, or uncountable.

  • Finite Sets are sets that have finitely many elements, is a finite set of cardinality 3.
  • Countable Sets are sets that have as many elements as the set of natural numbers, so the set of rational numbers is countable.
  • Uncountable Sets are sets that have more elements than the set of natural numbers, so the set of real numbers is uncountable.

Common Sets of Numbers[edit]

Diagram to demonstrate the number systems ℝ, ℚ, ℤ and ℕ as sub-sets of each other.
  • is the set of Naturals
  • is the set of Integers
  • is the set of Rationals
  • is the set of Complex Numbers

Partitions of Sets[edit]

In many applications of set theory, sets are divided up into non-overlapping (or disjoint) pieces. Such a division is called a partition.

Two sets are called disjoint if, and only if, they have no elements in common.

A and B are disjoint ⇔ A ∩ B = ∅.

Sets are mutually disjoint (or pairwise disjoint or nonoverlapping)

if, and only if, no two sets and with distinct subscripts have any elements in

common. More precisely, for all

whenever .

Power Sets[edit]

The power set of a set A is all possible subsets of A, including A itself and the empty set. Which can be represented:

For the set

Learning resources[edit]

Wikiversity[edit]

Wikipedia[edit]