SET THEORY / Teori Himpunan

SET (HIMPUNAN)

A. The Definition

 Set is collection or group of things that can be defined (well defined) and distinct objects.

 

A set cab be written in curly brackets {...} as a symbol of set or using the word "the set of". Sets can also be denoted using capital letter as name of the set.

B. Elements Of Set 

Elements are the objects conteined in a set. A set may be defined by a common property amost the objects. The element is doneted by the symbol "∈". The negation of set membership is denoted by the symbol "∉" (not element).

Example
1, 3, 5, 7 ∈ {odd numbers}
2 ∉ {odd numbers}
0, 2 ∉ {odd numbers}

C. Cardinality Of Sets

The number of elements in a particular set is a property known as cardinality; informally, This is the size of a set. The cardinality of a set A, denoted n(A) or |A|,

Example:
E = {even number less than 10}
E = {2, 4, 6, 8}
n(E) = 4

Three type of cardinality of set:

1. Empty Set

   The cardinality of the empty set is zero, is denoted by the symbol { } or Ø

    Example:
    P = {prime number less than 2} → P = { }
    n(P) = 0

2. Finite Set

    An finite set is a set with a finite number of elements.

    Example:

    G = {odd number less than 10} → G = {1, 3, 5, 7, 9}

    n(G) = 5

3. Infinte Set

    An infinite set is a set with an infinite number of elements.

    Example:

    A = {whole number}  → A = {0, 1, 2, 3, ...}

    n(A) = ~ (infinite symbol)

D. Represent A Set 

There are three common ways of representing a set.

1. Using words

    Example:
    - The set of vowels

    - {the color of rainbow}

    - {the day names starting with t}

2. Using roster notation (tabular form or list the elements)

    The Roster notation (or enumeration notation) method of defining a set consist of listing each            element or member of the set.

    Example:

    - {2, 3, 5, 7}

3. Using set-builder notation

    Example:

    - {y | 0 ≤ y < 10 , y ∈ whole number}

 

   Using word                           : The set of prime number between 1 and 10

   Using roster notation            : {2, 3, 5, 7}

   Using set-builder notation    : {x | 1 < x < 10, x ∈ prime number}

E. Special Set

1) Z or ℤ, denoting the set of all integers (whether positive, negative or zero): 

    Z = {..., −2, −1, 0, 1, 2, ...}

2) N or ℕ, denoting the set of all natural numbers: N = {1, 2, 3, ...} 

    (sometimes defined excluding 0)  

3) A, denoting the set of all whole numbers: A = {0, 1, 2, 3, 4, …}

4) G, denoting the set of all odd numbers: G = {1, 3, 5, 7, …}

5) E, denoting the set of all even numbers: E = {2, 4, 6, 8, …}

6) P or ℙ, denoting the set of all prime numbers: P = {2, 3, 5, 7, 11, 13, 17, ...}

7) T, denoting the set of all composite numbers: T = (4, 6, 8, 9, …}

For further explanation, can be seen in the following video: