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Set Theory

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.

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Set Theory

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  1. Set Theory Prepared by: Still John F. Reyes

  2. 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

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

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

  5. 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. • set builder notation – a modification of the rule method.

  6. Methods of Listing the Elements of a Set • Examples: • The symbol ( / ) means “wherein” or “such that.”

  7. Subset • 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

  8. Kinds of Sets • 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.

  9. Kinds of Sets • 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.

  10. 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.

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

  12. Set Operations • 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. ( - )

  13. Thank You for Listening!!!

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