Chapter-2-Symbols-and-SET-Theory.pdfffff
- 1. SYMBOLS and SET Theory 1 Prepared by: ARMANDO C. MANZANO
- 2. Sets Definition of a Set Methods of naming a set Properties of Sets Operation on Sets Venn Diagram Mathematical Language & Symbols 21
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- 4. Basics of Sets • Georg Cantor- introduced the study of sets • Naming a Set- use capital letter of the English Alphabet • Elements ∈- denote membership of a set • ∉ means “not an element” • Let A be set of PSUBC student • B is set of letters a b c ∈B but 5 ∉B C is a course of PSU Bayambang
- 6. Describing a Set 1. Word Description— • A be a set of PSUBC student • B is set of letters • C is a course of PSU Bayambang • D BPA student of PSU Bayambang • P is BPED student of PSUBC • 2. Listing Method or Roster Method- list all the elements of set, separated by commas and enclose by braces • Let R be colors of rainbow • 𝑅 = {𝑟𝑒𝑑, 𝑜𝑟𝑎𝑛𝑔𝑒, 𝑦𝑒𝑙𝑙𝑜𝑤, 𝑔𝑟𝑒𝑒𝑛, 𝑖𝑛𝑑𝑖𝑔𝑜, 𝑣𝑖𝑜𝑙𝑒𝑡, 𝑏𝑙𝑢𝑒} • C={BPA, BSBA, ICT, BSN, BSE, BEE, BPED, BTLED, BECED,ABEL}
- 7. 3. Rule Method or Set- builder form= using the description {x | x is ….} read as x such that x is… • 𝑅 = {𝑥|𝑥 𝑖𝑠 𝑐𝑜𝑙𝑜𝑟 𝑜𝑓 𝑟𝑎𝑖𝑛𝑏𝑜𝑤} • 𝐶 = {𝑥|𝑥 𝑖𝑠 𝑐𝑜𝑢𝑟𝑠𝑒 𝑖𝑛 𝑃𝑆𝑈 𝐵𝑎𝑦𝑎𝑚𝑏𝑎𝑛𝑔} • 𝐸 = {𝑥|𝑥 𝑖𝑠 𝐺𝑜𝑑𝑠 𝑜𝑓 𝐺𝑟𝑒𝑒𝑘 𝑀𝑦𝑡ℎ𝑜𝑙𝑜𝑔𝑦} • • Let 𝑁 = {𝑥|𝑥 𝑖𝑠 𝑙𝑒𝑡𝑡𝑒𝑟 𝑜𝑓 𝑦𝑜𝑢𝑟 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑛𝑎𝑚𝑒} use Roster method to describe • Piolo Ramsay Avelino • N={ A, P, I, O, L, M, E, N, S,V, Y, R} • 4. Modified Roster • 𝐸 = {𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟} in modified roster as 𝐸 = {2, 4, 6, 8, … } • 𝑂 = 𝑥 𝑥 𝑖𝑠 𝑡𝑤𝑜 𝑑𝑖𝑔𝑖𝑡 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 , then • 𝑂 = {11, 13, 15, 17, 19,21, … , 97, 99}
- 9. Kinds of Sets 1. Unit set- set contains exactly one element 2. Empty or Null Set- { } or ∅ Set contains no element at all, {∅} not null 3. Finite Set- set contains countable element 4. Infinite Set – set contains uncountable elements Cardinality of sets – number of elements in a given set Let 𝐴 given set then 𝑛(𝐴) is cardinality of set A 5. Universal Set U- totality of all elements under consideration Augustus De Morgan- he was the one who introduced universal set • Let 𝐴 = {𝑥|𝑥 𝑖𝑠 𝑟𝑎𝑖𝑛𝑏𝑜𝑤 𝑐𝑜𝑙𝑜𝑟} • 𝐵 = {𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟} • 𝐶 = {𝑥|𝑥 𝑖𝑠 𝑑𝑖𝑔𝑖𝑡 𝑜𝑓 𝐻𝑖𝑛𝑑𝑢 𝐴𝑟𝑎𝑏𝑖𝑐 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑠𝑦𝑠𝑡𝑒𝑚} • 𝑛 𝐴 = 7 finite • 𝑛 𝐵 = ∞, infinite • 𝑛 𝐶 = 10, finite
- 10. Set Relations 1. Joint Sets- sets contains at least one common element 2. Disjoint Sets- sets contains no common element at all 3. Equivalent Sets(∼)– sets contains same cardinal or number of elements 4. Equal Sets (=) – sets contains similar or common elements Note: All equal sets are equivalent but not all equivalent sets are equal
- 11. Subsets • Subsets- a set is a subset of any sets when all its elements contained on the sets • Let A, B and C is a given set and U as universal set • ⊂:proper subset – when a subset has a cardinality lesser the set given • ⊆:improper subset- a set itself and null set • ⊃ super set= the mother set *Null set is a subset of any set *The set itself is a subset of itself • • Let 𝑈 = {1, 2, 3, 4, 5, 6, 7, 8} • 𝐴 = {1, 𝟐, 𝟑, 𝟒} • 𝐵 = {3, 4, 5, 6} • 𝐶 = {1, 3, 5, 7} • 𝐷 = {4, 8} • We say now that 𝐴, 𝐵, 𝐶, 𝐷 ⊂ 𝑈 • While null set and U itself is an improper subset (U ⊆ U, ∅ ⊆ U)
- 12. Power Set = set contains all possible subsets of a given sets Let 𝐴 be a given set then ℘ 𝐴 is power set ofA • With respect to A= {1, 2, 3, 4} , we have subsets { 1 }, { 2}, { 3 }, {4}, {1, 2}, {1, 3}, {1, 4},{2, 3}, {2,4}, {3,4}, {1, 2,3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4} are proper subsets of A, then we also say that A is the super set of the sets • While ∅ and {1, 2, 3, 4} are improper subsets of A • Using the above example, let 𝐴 = {1, 2, 3, 4} then • ℘ 𝐴 ={{ 1 }, { 2}, { 3 }, {4}, {1, 2}, {1, 3}, {1, 4},{2, 3}, {2,4}, {3,4}, {1, 2,3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4} , { }} • 2^4=16 • {1,2,3}=2^3=8
- 13. • B= {𝑎, 𝑏, 𝑐}
- 14. Set Operation Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷 ⊂ 𝑈 1. Union of sets (∪) - set contains either elements of two given sets • 𝐴 ∪ 𝐵 = 𝑥 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 = 𝑥 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 2. Intersection of Sets (∩) - set contains common elements of the two given sets • 𝐴 ∩ 𝐵 = 𝑥 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 = 𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 3. Difference of Sets (–)- set contain elements not found on other given set • 𝐴– 𝐵 = 𝑥 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵 • 𝐵– 𝐴 = 𝑥 𝑥 ∉ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 4. Complement of a Set (′)- set contain elements not found on the given set but elements of the universal set • 𝐴′ = 𝑥 𝑥 ∉ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝑈 • 𝐵′ = 𝑥 𝑥 ∉ 𝐵 𝑎𝑛𝑑 𝑥 ∈ 𝑈
- 15. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 1. Complement of : A, B, C, D, E, F 𝐴′ = {𝑜, 𝑝, 𝑞, 𝑟} 2. Union: 𝐴 ∪ 𝐵, 𝐴 ∪ 𝐶, 𝐸 ∪ 𝐵, 𝐵 ∪ 𝐷, 𝐴 ∪ 𝐸 𝐶 ∪ 𝐵, 𝐸 ∪ 𝐷, 3. Intersection: 𝐴 ∩ 𝐵, 𝐶 ∩ 𝐵, 𝐴 ∩ 𝐸, 𝐷 ∩ 𝐵, 𝐶 ∩ 𝐸 𝐵 ∩ 𝐸 𝐷 ∩ 𝐸 4. Difference: 𝐴– 𝐵 , 𝐴– 𝐶, 𝐷– 𝐵, 𝐴– 𝐸, 𝐵– 𝐶, 𝐵– 𝐸, 𝐷– 𝐸
- 16. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 1. Complement of : A, B, C, D, E, F 𝐴′ = {𝑜, 𝑝, 𝑞, 𝑟}
- 17. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 2. Union: 𝐴 ∪ 𝐵, 𝐴 ∪ 𝐶, 𝐸 ∪ 𝐵, 𝐵 ∪ 𝐷, 𝐴 ∪ 𝐸 𝐶 ∪ 𝐵, 𝐸 ∪ 𝐷,
- 18. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 3. Intersection: 𝐴 ∩ 𝐵, 𝐶 ∩ 𝐵, 𝐴 ∩ 𝐸, 𝐷 ∩ 𝐵, 𝐶 ∩ 𝐸 𝐵 ∩ 𝐸 𝐷 ∩ 𝐸
- 19. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 4. Difference: 𝐴– 𝐵 , 𝐴– 𝐶, 𝐷– 𝐵, 𝐴– 𝐸, 𝐵– 𝐶, 𝐵– 𝐸, 𝐷– 𝐸
- 20. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 1. Complement of : A, B, C, D, E, F 𝐴′ = {𝑜, 𝑝, 𝑞, 𝑟}
- 21. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 2. Union: 𝐴 ∪ 𝐵, 𝐴 ∪ 𝐶, 𝐸 ∪ 𝐵, 𝐵 ∪ 𝐷, 𝐴 ∪ 𝐸 𝐶 ∪ 𝐵, 𝐸 ∪ 𝐷,
- 22. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 3. Intersection: 𝐴 ∩ 𝐵, 𝐶 ∩ 𝐵, 𝐴 ∩ 𝐸, 𝐷 ∩ 𝐵, 𝐶 ∩ 𝐸 𝐵 ∩ 𝐸 𝐷 ∩ 𝐸
- 23. Exercise: Let U be universal set and 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 ⊂ 𝑈 Given: 𝑈 = {𝑘, 𝑙, 𝑚, 𝑛, 𝑜, 𝑝, 𝑞, 𝑟} 𝐴 = {𝑘, 𝑙, 𝑚, 𝑛} 𝐵 = {𝑚, 𝑛, 𝑜, 𝑝} 𝐶 = {𝑘, 𝑚, 𝑜, 𝑞} 𝐷 = {𝑙, 𝑛, 𝑝, 𝑟} 𝐸 = {𝑘, 𝑛, 𝑟} 𝐹 = {𝑚, 𝑜} Find 4. Difference: 𝐴– 𝐵 , 𝐴– 𝐶, 𝐷– 𝐵, 𝐴– 𝐸, 𝐵– 𝐶, 𝐵– 𝐸, 𝐷– 𝐸
- 25. Venn Diagram • Venn Diagram- is a set diagram introduced by John Venn in 1880. • He invented it to illustrate relations between a finite collection of sets. It is also often used to illustrate set operations.
- 28. • AVenn diagram is a diagram representing mathematical or logical sets pictorially as circles within an enclosing rectangle (the universal set), common elements of the sets being represented by the areas of overlap among the circles.
- 29. Venn on Operation of Sets
- 30. Some Illustrations on Operation on Sets
- 31. Solve The Problem using a Venn Diagram A history teacher was interested to know about her class of 42 students who keeps up with current events. She gathered the following data: 9 students read the newspaper, 18 students listen to the radio, 30 students watch television, 3 students both read newspaper and listen to the radio, 12, students both listen to the radio and watch television, 6 students both read the newspaper and watch television, and 2 students read the newspaper, listen to the radio and watch television. Organize the data using the Venn Diagram.
- 32. 9 students read the newspaper, • 18 students listen to the radio, • 30 students watch television, • 3 students both read newspaper and listen to the radio, • 12, students both listen to the radio and watch television, • 6 students both read the newspaper and watch television, and • 2 students read the newspaper, listen to the radio and watch television.
- 33. The problem will be illustrated by working backward or starting fro the bottom
- 34. Form the given Venn diagram below • Form the given Venn diagram below • How many students a. use exactly one source of information b. ate least two sources c. either watch tv or listen to a radio d. never use the given sources e. either read news paper or watch tv but not listening to a radio f. both watch tv and listen to a radio but not reading newspaper
- 35. Form the given Venn diagram below • Form the given Venn diagram below • How many students a. use exactly one source of information b. ate least two sources c. either watch tv or listen to a radio d. never use the given sources e. either read news paper or watch tv but not listening to a radio f. both watch tv and listen to a radio but not reading newspaper
- 36. Operations (Unary orBinary) Unary Operation is an operation on a single element. Example: negative of 5 multiplicative inverse of 7 Binary Operation is an operation that combines two elements of a set to give a single element. e.g. multiplication 3 x4 =12 Mathematical Language &Symbols 20
- 37. Properties of Real Numbers (Binary Operations) Let a, b, c and d be Real numbers 1. Closure Property 2. Commutative Property 3. Associative Property 4. Distributive Property 5. Identity 6. Inverse Note: Refer to your book page 28-30 Mathematical Language &Symbols 20
- 38. Subsets Real numbers 1. Natural number/counting N={1,2,3,4,… 2. Whole number W={0, 1,2 ,3, 4,…} 3. Integers(Z):0, Z+=N, Z-: Z={0, 1,-1, 2, -2,…} 4. Rational(Q): { 𝑎 𝑏 𝑎, 𝑏 ∈ 𝑍, 𝑏 ≠ 𝑜 , fractions and decimal 5. Irrationals (Q’) Mathematical Language &Symbols 20
- 39. Properties of Real Numbers (Binary Operations) Let a, b, c and d be Real numbers 1. Closure Property a+b= real number a(b)= real number 2. Commutative Property a+b=b+a axb=bxa Mathematical Language &Symbols 20
- 40. Properties of Real Numbers (Binary Operations) 3. Associative Property a+(b+c)=(a+b)+c=(a+c)+b a(b.c)=(a.b)c=(a.c)b 4.Distributive Property a(b+c)=a(b)+a(c) (a+b)c=a(c)+ b( c) 5. Identity a+0=a 5+___=5 a(1) =a 5x___=5 6.Inverse a+(-a) =0 5+___=0 a(1/a) =1 5x___=1 Mathematical Language &Symbols 20
- 41. Properties of Real Numbers (Binary Operations) 3. Associative Property a+(b+c)=(a+b)+c=(a+c)+b 2+(3+4)=(2+3)+4=(2+4)+3 a(b.c)=(a.b)c=(a.c)b 2(3.4)=(2.3)4=(2.4)3 4.Distributive Property a(b+c)=a(b)+a(c) left hand (a+b)c=a(c)+b(c) right hand 5. Identity a+0=a 5+___=5 a(1)=a 5x___=5 6.Inverse a+(-a) =0 5+___=0 a(1/a) =1 5x___=1 Mathematical Language &Symbols 20