Set Theory Prepared by Still John F Reyes

Set Theory Prepared by: Still John F. Reyes

Set Notation Set- is a collection or aggregate of definite, distinct objects. A well-defined set means that it is possible to determine whether an object belongs to a given set. Elements of a Set- the objects or members of a set. • Symbol: ε (epsilon)- use to denote the element of a set. Ex: r ε A ε – use to denote that an element is not an element of the given set. Ex: b ε A

Set Notation Ex. of well-defined sets (defined) 1. Set of ace cards 2. Colors of the rainbow 3. Days of the week Ex. of not well-defined sets (undefined) 1. Set of cards 2. Set of books 3. Set of beautiful women in Asia

Set Notation Different symbols are used when dealing with sets: 1. A pair of braces { } – is used to represent the idea of a set. 2. Capital letters of the English alphabet – are used to name sets. Example: A = { a, b, c, d, e } B = { 2, 4, 6, 8, 10 }

Methods of Listing the Elements of a Set 1. Roster or Tabular Method – listing all the elements, enclosed it in braces and separated by comma. Ex: C = { a, b, c } B = { 1, 2, 3, 4, 5 } 2. Rule Method – a conditional way of listing method by writing and description using a particular variable. v set builder notation – a modification of the rule method.

Methods of Listing the Elements of a Set Examples: Rule Set-builder 1. A = { counting numbers less than 5} A = { x/x is a counting number less than 5 } 2. B = { days of the week that begin with letter S } B = { x/x is a day of the week beginning with letter S } v The symbol ( / ) means “wherein” or “such that. ”

Subset – it is a part of a given set. Two kinds of subset: Proper Subset – a part of a set, symbol ( ⊂ ) Improper Subset – the given set is equal to that set, symbol ( ⊆ ) Number of Subsets of a Given Set: - If a set contains n number of elements then the number of subsets is 2ⁿ

Kinds of Sets 1. 2. 3. 4. Empty or Null Set – sets having no elements. Symbol: { } or ∅ Universal Set (U) – also called the general set, is the sum of all sets or the totality of elements under consideration or a particular discussion. Unit Set – set having only one element. Finite Set – sets having a limited or countable number of elements.

Kinds of Sets 5. 6. 7. 8. 9. Infinite Set – sets having an unlimited or uncountable number of elements. Equal Sets – sets having the same elements. Symbol: (=) Equivalent Sets – sets having the same number of elements. Symbol: (~) Joint Sets – sets that have elements in common. Disjoint Sets – sets that have no elements in common.

Venn Diagram It is a graphical representation, usually circular in nature. It is one way of showing the relationships of two or more sets by the use of pictures. This method was developed by John Venn (1834 -1883) thus, the name Venn Diagram. It consists of a rectangle representing the universal set and circles that represent the sets. Sometimes, circles can also represent the universal set.

Venn Diagram U A B a, b, c d, e, f

Set Operations 1. 2. 3. 4. Union – it shows the unity of two or more sets. It is the joining of sets. ( ∪ ) Intersection – it shows the intersection of the common elements of sets. ( ∩ ) Complement of a Set – it is the set whose elements are in the universal set but not in a set or a given set. ( ʼ ) Difference – set of elements found in a set but not belong or found in other set. ( - )

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