1. Set Notation and Description Kinds of Sets Operations on Sets By: Mr. Gerzon B. Mascariñas

2. GBM 2012 What is set? A set is a collection of well-defined objects or things. Capital letter is used to name a set. An element is an object contained in a set Example: A = { 2, 4, 6, 8} 2, 4, 6, and 8 are called elements of set A.

3. GBM 2012 Set Notation Elements an element is a member of a set notation:  means “is an element of”  means “is not an element of” Examples: –A = {1, 2, 3, 4} 1  A6  A 2  Az  A –B = {x | x is an even number  10} 2  B9  B 4  Bz  B

4. GBM 2012 Set Theory Notation SymbolMeaning Upper casedesignates set name Lower casedesignates set elements { }enclose elements in set  or is (or is not) an element of  is a subset of (includes equal sets)  is a proper subset of  is not a subset of  is a superset of / or :such that (if a condition is true) | |the cardinality of a set

5. GBM 2012 Ways of Describing a Set The roster or listing method: –A = {2,4,6,8,10} Descriptive method - Set A is a set of positive even integers less than 12. Set builder notation method –A = {x|x is a positive even integer less than 12} –Read as set A is a set of all X such that X is a positive even integer less than 12

6. Describe the following: S = { Math, English, Filipino, Science} Descriptive method - Set S is a set of major subjects. Set builder notation method - S = { x/x is a major subject} GBM 2012

7. L = {kilogram, hectogram, gram, milligram} Descriptive method - Set L is a set of metric unit measures of weight. Set builder notation method - L = { x/x is a metric unit measures of weight.} GBM 2012

8. Identify the elements of the following: 1. Set R is a set of prime numbers less than 21. - R = { 2, 3, 5, 7, 11, 13, 17, 19 } 2. Set G is a set of factors of 36. - G = { 1, 2, 3, 4, 6, 9, 12 } 3. Set H is a set of vowels of the English alphabet. - H = { a, e, i, o, u } GBM 2012

9. 4. Set Y is a set of even numbers greater than 6 but less than 10. - Y = { 8 } 5. Set M is set of months of the year beginning in M. - M = { March, May } GBM 2012

10. KINDS OF SETS 1. Finite set – a set that has last element or it has a countable number of elements Example: A = {x/x is a prime number less than 30} The elements of set A can be counted. There are 10 elements namely 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. GBM 2012

11. Finite Set Cardinality Cardinality refers to the number of elements in a set Set Definition Cardinality A = {x/x is a lower case letter} |A| = 26 B = {2, 3, 4, 5, 6, 7} |B| = 6 C = {x/x is an even number  10} |C|= 4 D = {x/x is an even number  10} |D| = 5

12. GBM 2012 2. Infinite set – a set with unlimited number of elements or it has at least as many elements as the set of natural numbers. In listing the elements of an infinite set, we used ellipses (…). This indicates that there are still many elements that follow.

13. Example: Set R is a set of even numbers. R = { 2, 4, 6, 8,... } B = {x/x is a whole number} B = {1, 2, 3, 4, 5, 6, 7,... } D = {x/x is a multiple of 10} D = {10, 20, 30, 40, 50, 60,...} GBM 2012

14. Infinite Set Cardinality Set DefinitionCardinality A = {1, 2, 3, …}|A| = B = {x | x is a point on a line}|B| = C = {x| x is a point in a plane} |C| =

15. GBM 2012 3. Equal set - sets that contain precisely the same elements and same number of elements. The order in which the elements are listed is unimportant. Examples: A = {1, 2, 3, 4} B = {1, 4, 2, 3} A  B and B  A; therefore, A = B and B = A A = {c, a, r, e, s} B = {a, r, c, s, e} A  B and B  A; therefore, A = B and B = A

16. GBM 2012 4. Equivalent set - sets that have the same number of elements. Examples: A = {1, 2, 3, 4} B = {p, o, n, d} Set A has 4 elements, Set B has 4 elements therefore, A B and B A D = {j, u, n, e} E = {j, u, l, y} Set D has 4 elements, Set E has 4 elements therefore, A B and B A

17. GBM 2012 5. Universal set is a set contains all the elements that the other sets have - it is denoted by the symbol U Example: U = {all students at VPS} Some Subsets: A = {all Kinder students} B = {Grade school students} C = {Junior high school students} D = {Senior high school students}

18. Given: A = {faith, service, love, justice, learning} B = { } C = {justice} D = {love, faith, service} Set A is a universal set. Set C and D are subsets of A because their elements can be found in set A. Set B has no element. It is called an empty set. An empty set is a subset of any set. GBM 2012

19. 6. Empty set or Null set – a set without an element. - it is represented by { } or . 7. Subsets - exists when a set’s members are also contained in another set  means “is a subset of”  means “is a proper subset of”  means “is not a subset of” GBM 2012

20. Subset Relationships A = {x | x is a positive integer  8} set A contains: 1, 2, 3, 4, 5, 6, 7, 8 B = {x | x is a positive even integer  10} set B contains: 2, 4, 6, 8 C = {2, 4, 6, 8, 10} set C contains: 2, 4, 6, 8, 10 Subset Relationships A  AA  BA  C B  AB  BB  C C  AC  BC  C

21. GBM 2012 The Power Set ( P ) The power set is the set of all subsets that can be created from a given set The cardinality of the power set is 2 to the power of the given set’s cardinality notation: P ( set name) Example: A = {a, b, c}where |A| = 3 P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A,  } and | P (A)| = 8 In general, if |A| = n, then | P (A) | = 2 n

22. GBM 2012 Special Sets Z represents the set of integers –Z + is the set of positive integers and –Z - is the set of negative integers N represents the set of natural numbers n ℝ represents the set of real numbers Q represents the set of rational numbers

23. Operations of Sets 1. Union of Sets – denoted by U. To get the union of the sets, we just put together the elements of the sets without repeating any of the element. Given:A = {20, 40, 60, 80} B = {10, 30, 50, 70} C = {10, 20, 30, 40} A U B = {10, 20, 30, 40, 50, 60, 70, 80} A U C = {10, 20, 30, 40, 60, 80} GBM 2012

24. 2. Intersection of sets – denoted by ∩ - the common element/s between the given sets Given:A = {20, 40, 60, 80} B = {10, 30, 50, 70} C = {10, 20, 30, 40} A ∩ B = { } B ∩ C = {10, 30} A ∩ B ∩ C = { } GBM 2012

25. 3. Complements of Sets Denoted by the symbol (‘) A set of elements which can be found in the universal set but not in the given set. Given: U = { 0, 1, 2,... 10} A= {0, 1, 2, 3, 4, 5} B= {1, 3, 5, 7, 9} Find: A’ = {6, 7, 8, 9, 10} B’ = {0, 2, 4, 6, 8, 10} (A’)’ = { } GBM 2012

26. Exercises: GBM 2012

27. Venn Diagrams Venn diagrams show relationships between sets and their elements Universal Set Sets A & B

28. GBM 2012 Venn Diagram Example 1 Set DefinitionElements A = {x | x  Z + and x  8}1 2 3 4 5 6 7 8 B = {x | x  Z + ; x is even and  10}2 4 6 8 10 A  B B  A

29. GBM 2012 Venn Diagram Example 2 Set DefinitionElements A = {x | x  Z + and x  9} 1 2 3 4 5 6 7 8 9 B = {x | x  Z + ; x is even and  8}2 4 6 8 A  B B  A A  B

30. GBM 2012 Venn Diagram Example 3 Set DefinitionElements A = {x | x  Z + ; x is even and  10} 2 4 6 8 10 B = x  Z + ; x is odd and x  10 } 1 3 5 7 9 A  B B  A

31. GBM 2012 Venn Diagram Example 4 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} A = {1, 2, 6, 7}

32. GBM 2012 Venn Diagram Example 5 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7} B = {2, 3, 4, 7}

33. GBM 2012 Venn Diagram Example 6 Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 6, 7} B = {2, 3, 4, 7} C = {4, 5, 6, 7}

34. Problem Solving 200 people entered a carnival. 60 people tried the Octopus, 100 people tried the Ferris Wheel while 40 people tried both the Octopus and the Ferris Wheel. a. How many tried the Octopus only. b. How many tried the Ferris Wheel only? c. How many tried the other rides? GBM 2012

35. Set U = 200 people Let set O = people tried Octopus Let set F = people tried Ferris wheel Let set R = people tried other rides O F 20 40 60 R 80 GBM 2012

36. The following information was obtained out of 100 grade 7 students. 50 like dancing 41 like singing 38 like acting 10 like dancing, singing, and acting 20 like dancing and singing 25 like singing and acting 15 like dancing and acting GBM 2012

37. How many students do not like the three activities? How many students like dancing only? How many students like singing only? How many students like acting only? How many students like acting and singing only? How many students like singing and dancing only? How many students like singing and acting only? GBM 2012

38. A survey result showed that 150 pupils love to read books during their free time and 80 pupils love to play chess. If 70 pupils love both reading and playing chess and 40 pupils do other things, how many pupils participated in the survey? GBM 2012