# Concepts of Set theory relevant to Probability

# Set

A set is a collection of distinct objects considered as a whole.

- {1, 2, 3, 4, 5, ... ∞ }
- {Cat, Dog, Horse, Bull, Tiger}
- {α, β, ♥, γ, ε ♣ }

## Denoting a Set

Sets are generally denoted by the Capital Letters of English alphabets.

- A = {1, 2, 3, 4, 5, ... ∞ }
- Ω = {Cat, Dog, Horse, Bull, Tiger}
- M = {α, β, ♥, γ, ε ♣ }

# Elements of a Set

### Element

- A small part of the total whole.
- An distinct part of a composite entity that can be separated and re-attached.
- Component
- Constituent
- Ingredient

Any identifiable entity may form the element of a set.

**Eg** : Numbers, Letters, Points, Birds, Symbols, Things, Equations, Persons etc.

A set can also be an element of another set.

N = { {1, 2, 3, ... 10} , {11, 12, 13, ... 20}, {21, 22, 23, ... 30} }

## Order of Elements

The elements of the set may be listed in any order. What the elements are is the only thing that matters.# Describing Sets

## Intensional/Implicit Definition

### Intension

- What one must know in order to determine the reference of an expression
- An idea that is implied or suggested
- Connotation

### Implicit

- Implied though not directly expressed;
- Being without doubt or reserve
- inexplicit
- unquestioning

Intention is different from intension.

### Intention

- An anticipated outcome that is intended or that guides planned actions
- aim
- purpose

Intensional/Implicit definition of a set involves defining the set in terms of the necessary and sufficient conditions required to be satisfied for an element to belong to the set.

**A**is the set of even numbers divisible by 4.Commonly written as

**A**= {Even numbers divisible by 4}**M**- The set of all models of cars in a city**M**= {Models of cars in a city}**G**- The set of all democratic countries in South Asia**G**= {Democratic countries in South Asia}

The braces "{ }" replace the words "The set of all".

## Set Builder Notation

The set builder notation uses a variable to represent the elements of the set along with the necessary and sufficient conditions required to be satisfied for an element to belong to the set.

**P**= { x : x ∈ Z, − 100 ≤ x ≤ 100 and x ÷ 5}**V**= { x^{2}− 3x + 4 | x ∈ N, x ≤ 12}

The symbol ":" is read "such that"

## Extensional/Explicit/Enumerative Definition

### Extensional

- defining a word by listing the class of entities to which the word correctly applies
- denotive

### Explicit

- Precisely and clearly communicated or readily observable; leaving nothing to implication
- denotative
- expressed

### Enumerative

- Counting, or reckoning up, one by one

The extensional or explicit or enumerative definition of a set involves listing out all the elements that belong to that set. Such a description/definition of the set is possible/practical only where the elements of the set are finite/small.

**M**= {Toyota Corolla, Honda City, Benz C Class, Ford Escape, Nissan Maxima}**G**= {USA, Mexico, Cuba, Panama, Canada, Colombia}

## Ostensive/Abbreviated Definition

### Ostensive

- Represented or appearing as such
- Manifestly demonstrative

### Abbreviate

- Reduce in scope while retaining essential elements
- Shorten

The ostensive or abbreviated definition of a set involves listing out only some of the elements of the set (but not all) to create an understanding on the nature of the elements within the set.

**A**= {2, 4, 6, 8, ... }**T**= {5, 10, 15, ... 245, 250}

## Belonging and Not Belonging to a Set

The elements which satisfy the necessary and sufficient conditions in relation to a set are said to belong to the set and those that do not satisfy the conditions are said to be not belonging to the set.Where A = {Even numbers less than 50 divisible by 4}

- 16 ∈ A
Read as 16 belongs to set A

- 35 ∉ A
Read as 35 does not belong to set A

# Cardinality of a Set

It is denoted by the name of the set included within two vertical bars as |D| or within parentheses following n as n(D) indicating the number of elements in the set D.

- A = {5, 10, 15, 20, ... , 245, 250}
⇒ n(A) = 50

- Ω = {α, β, ♥, γ, ε ♣ }
⇒ n(Ω) = 6

- A = {a, e, i, o, u};
⇒ n(A) = 5

- B = {a, b, c, d, e, f, ...., w, x, y, z}
⇒ n(B) = 26

- N = {1, 2, 3, ....} ⇒
n(N) = ℵ

_{0}read aleph nought

- A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}
⇒ n(A ∪ B) = 8

- K ∩ L ∩ M = {a, o}
⇒ n(K ∩ L ∩ M) = 2

- B − A = {b, c, d, f}
⇒ n(B − A) = 4

- F = { {1, 2, 3, ... 10} , {11, 12, 13, ... 20}, {21, 22, 23, ... 30} }
⇒ | F | = 3

Each element of the set F is a set.

Since there are three sets which form elements of the set F, the cardinality of set F is 3. The elements of those sets should not be counted in finding the cardinality of F.

## Empty/Null Set

A set whose cardinality is Zero i.e. a set without any elements is an empty or null set.It is represented as {} or by the symbol Φ

M = {Odd number divisible by 2}

⇒ M = Φ or M = {}

## M = Φ ⇏ M = {Φ}

Whereas M = Φ represents a Null set, M = {Φ} is a set with only one element ΦThis can be better understood if they are represented as M = {} and M = {Φ}

## Sets with infinite Cardinality

To express the cardinality of a set, the number of elements in the set should be countable.We use the term ∞ to express the boundaries of of some common number sets.

- Set of Natural numbers
N = {1, 2, 3, ... ∞ }

- Set of Whole numbers
W = {0, 1, 2, 3, ... ∞ }

- Set of Integers
Z = {− ∞, ... − 3, − 2, − 1, 0, 1, 2, 3, ... ∞}

- Set of Real Numbers
R = {− ∞, ... − 2, − 3/2, − 1, 0, 1, 3/2, 2, ... ∞ }

The infinite (∞) symbol here is used to define the boundaries and not the infiniteness of the cardinality of these sets.

## Finite, Countably infinite, Uncountable

The Cardinality of the set of Natural numbers is represented by ℵ_{0}(read aleph null).

## Finite sets

Any set with a cardinality less than the cardinality of the set of Natural numbers is a finite set. A set P is said to be finite if n(P) < n(N) or |P| < ℵ_{0}

## Countably infinite sets

Any set with a cardinality equal to the cardinality of the set of Natural numbers is a countably infinite set. A set M is said to be finite if n(M) = n(N) or |M| = ℵ_{0}

## Uncountable sets

Real numbers set is also called continuum. The Cardinality of the continuum (Real number set) is represented by **C**.

The cardinality of continuum is greater than the cardinality of the natural number set i.e. **C** > ℵ_{0}

Any set with a cardinality greater than the cardinality of the set of Natural numbers is said to be an uncountable set.

A set S is said to be uncountable if n(S) > n(N) or |S| > ℵ_{0}

# Union of Sets

The symbol **∪** is used to indicate union.

- A = {1, 2, 3, 4, 5};
B = {2, 4, 6, 8, 10}

⇒ A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}

A ∪ B = {x / x ∈ A or x ∈ B}

- K = {a, e, i, o, u};
L = {a, b, c, d, e, o};

M = {a, o, p, q, r, s, t, u}

⇒ K ∪ L ∪ M = {a, b, c, d, e, i, o, p, q, r, s, t, u}

K ∪ L ∪ M = {x / x ∈ K or x ∈ L or x ∈ M}

## Properties of union of sets

### Commutative Law

For any two sets A and BA ∪ B = B ∪ A

For any three sets A, B and C

A ∪ B ∪ C = B ∪ A ∪ C = B ∪ C ∪ A

### Associative Law

For any three sets A, B and CA ∪ (B ∪ C) = (A ∪ B) ∪ C

### Identity Law

A ∪ A = A### Complements Law

A ∪ Φ = A# Intersection of Sets

The symbol **∩** is used to indicate intersection.

- A = {1, 2, 3, 4, 5};
B = {2, 4, 6, 8, 10}

⇒ A ∩ B = {2, 4}

A ∩ B = {x / x ∈ A and x ∈ B}

- K = {a, e, i, o, u};
L = {a, b, c, d, e, o};

M = {a, o, p, q, r, s, t, u}

⇒ K ∩ L ∩ M = {a, o}

K ∩ L ∩ M = {x / x ∈ K and x ∈ L and x ∈ M}

## Disjoint sets

Two or more sets are said to be disjoint if they have no element in common i.e. if their intersection set is an empty set (a null set).

- A = {1, 3, 5, 7, ... };
B = {2, 4, 6, 8, 10, ...}

⇒ A ∩ B = {} Or A ∩ B = Φ

- K = {a, e, i, o, u};
L = {a, b, c, d, e};

M = {p, q, r, s, t, u}

⇒ K ∩ L ∩ M = {} (Or) K ∩ L ∩ M = Φ

## Properties of intersection of sets

### Commutative Law

For any two sets A and BA ∩ B = B ∩ A

For any three sets A, B and C

A ∩ B ∩ C = B ∩ A ∩ C = B ∩ C ∩ A

### Associative Law

For any three sets A, B and CA ∩ (B ∩ C) = (A ∩ B) ∩ C

### Identity Law

A ∩ A = A### Complements Law

A ∩ Φ = Φ# Complements of Sets

**U**.

Complements of sets are of two types.

## Relative Complements of Sets

Relative complement of a set in relation to a second set is the set of elements which are present only in the second set but not in the first.Where A and B are two sets, the relative complement of A in B is the set theoretic difference of B and A i.e. the set of elements present in B but not in A. It is written as B − A

Where

A = {a, e, i, o, u};

B = {a, b, c, d, e, f, ...., w, x, y, z}

Complement of A in B

= | B − A |

= | {a, b, c, d, e, f, ...., w, x, y, z} − {a, e, i, o, u} |

= | {b, c, d, f, ...., w, x, y, z} |

⇒ B − A = {x / x ∈ B and x ∉ A}

It is not a requirement that all the elements of the first set should be present in the second i.e. the first set need not be a subset of the second.

Where

A = {a, e, i, o, u};

B = {a, b, c, d, e, f}

Complement of A in B

= | B − A |

= | {a, b, c, d, e, f} − {a, e, i, o, u} |

= | {b, c, d, f} |

Some relevant properties of relative complements

- A − A = Φ
- A − Φ = A
- Φ − A = Φ

## Absolute Complements of Sets

For a set which is a subset of a universal set, the absolute complement is the relative complement of the set in the Universe. It is the set of all the elements in the universal set that are not present in it. The absolute complement of a set A is the relative complement of A in U (where U is the universal set). It is represented as A^{c} or A' [A^{c} or A' = U − A]

Where

U = {1, 2, 3, 4, 5, 6, ... };

O = {1, 3, 5, 6, 7, .... }

O^{c} | = | U − O |

= | {1, 2, 3, 4, 5, 6, ... } − {1, 3, 5, 7, 9, ...} | |

= | {2, 4, 6, 8, 10, ...} |

Some relevant properties of absolute complements

- A ∪ A
^{c}= A - A ∩ A
^{c}= Φ - Φ
^{c}= U - U
^{c}= Φ

## Absolute Relative Complements

For two sets A and B- A − B = A ∩ B
^{c} - (A − B)
^{c}= A^{c}∪ B

# Distributive Laws

For three sets A, B and C

- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

# De Morgan's Laws

- n(A ∪ B) = n (A) + n (B) − n(A ∩ B)
- (A ∪ B)
^{c}= A^{c}∩ B^{c} - (A ∩ B)
^{c}= A^{c}∪ B^{c}

n(A ∪ B ∪ C) = n (A) + n (B) + n (C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)