Lesson 3 - Language of Set PDF

Mathematics in the Modern World Lesson 3: The Language of Set Ms. Ana Mae B. Sinoy Professor  It is usually represented by CAPITAL LETTERS. Example: A = {a, e, i, o, u} ❑ Sets are denoted by braces and com...

Mathematics in the Modern World Lesson 3: The Language of Set Ms. Ana Mae B. Sinoy Professor  It is usually represented by CAPITAL LETTERS. Example: A = {a, e, i, o, u} ❑ Sets are denoted by braces and commas. A = {a, e, i, o, u} ❑ If an object belongs to the set, is called a member or element of the set denoted by the symbol ∈. Example: A = {a, e, i, o, u} Set A is the set of vowels in the English alphabet. a ∈ 𝑨 − 𝑟𝑒𝑎𝑑 𝑎𝑠 "𝑎 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝐴" Example: B = {even numbers from 1 to 10} B = {2, 4, 6, 8, 10} 6𝝐B 2𝝐 B 10 𝝐 B 1∉B 7∉B 12 ∉ B ❑ A set is said to be well-defined set if we know exactly if an object is an element of the set. Example: B = { whole numbers less than 5} B = { 0, 1, 2, 3, 4 } Example: D = { Natural numbers from 1 - 10} D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } E = { Days in a week} E = { M, T, W, Th, F, S, S } Example of not well - defined: F = { handsome guy in the school} G = { 3 favorite movies in 2022} H = { all positive integers} ❑ A variation of notation is used to used to describe a very large set. Example: { 1, 2, 3, ……50}Set of integers from 1 to 50 o The symbol.... is called ellipses and read as “and so forth”. 3 Ways in Describing a SET Listing Method/ Roster Method Rule Method/ Set-Builder Notation Interval Notation 3 Ways in Describing a SET 1.Listing Method/ Roster Method a set by listing the elements and enclosing them in braces. EXAMPLE: A = {a, e, i, o, u} - set of vowels from English alphabet B = {2, 4} - set of even numbers less than 5 C = {L, O, V, E} - set of letters from the word love. 3 Ways in Describing a SET 2.Rule Method/ Set-Builder Notation it indicate a set by enclosing in braces a descriptive phrase. used to define the elements of the set. EXAMPLE: Read as “ A is the set of all x such that x is A = { even numbers less than 5} even numbers less or A = {x/even numbers less than 5} than 5. Other Example: - set of x is all element of real numbers such { x 𝓔 ℝ /-3 < x < 6} that x is greater than -3 but less than 6 What are the elements of this set? {-2, -1, 0, 1, 2, 3, 4, 5} { x 𝓔 ℝ /-3 < x < 6} Open circle or hallow circle means Not included in the set Other Example: - set of x is all element of real numbers such { x 𝓔 ℝ /-3 ≤ x ≤ 6} that x is greater than or equal to -3 but less than or equal to 6 What are the elements of this set? {-3,-2, -1, 0, 1, 2, 3, 4, 5, 6} { x 𝓔 ℝ /-3 ≤ x ≤ 6} {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6} Closed or solid circle means included in the set 3 Ways in Describing a SET 3.Interval Notation 3 Ways in Describing a SET 3.Interval Notation 1. For a and b: [ ] means “included” ( ) means “ not included” 2. For ± ∞ : ( or) > or < 3 Ways in Describing a SET 3.Interval Notation Examples: [ 4, 8 ] = { 4, 5, 6, 7, 8 } ( 4, 8 ] = { 5, 6, 7, 8 } [ 4, 8 ) = { 4, 5, 6, 7 } ( 4, 8 ) = { 5, 6, 7 }  Cardinality of a Set Cardinality refers to the number of element in a set. We can denote the cardinality of set A by /A/ or n(A). Examples: Cardinality F= { 1, 2, 3, 4, 5 } n(F) = 5 C= { } n(C) = 0 ❑ A set with no members is called empty set or null set. It is denoted by the symbol { } or the Danish letter ∅. Examples: H = { positive whole number less than 0} H = { } or ∅ I = { triangle with 4 sides } I = { } or ∅  Finite Set A set is finite if it is possible to write down a complete list of all elements of the set and counting the number of elements. Examples: J = { 1, 3, 5, 7} H = { colors of the rainbow}  Infinite Set A set is infinite if it is not possible to write down a complete list of all elements of the set and counting the number of elements never to an end. Examples: L = { 0, 1, 2, 3, ….} M = { x/x is an integer} Equal and Equivalent Set Two sets are equal if they have exactly same elements. Examples: A = { 1, 3, 4} B = { 4, 3, 1} S = { 1, 3, 5} A = B, because they have the same elements. A ≠ S, because there elements are not the same.  Equal and Equivalent Set Two sets are equivalent if they have the same number of elements. Examples: K = { 5, 6, 8} C = { 2, 4, 6, 8} K is equivalent to F , because they have the same number of elements. K and C are not equivalent sets , because they do not have the same number of elements.  Let’s try! Determine if a given set is a well – defined set or not well – defined set. w 1. The set of cities in Metro Manila. w 2. The set of Math teachers in a particular school. nw 3. The set of books well – liked by my classmates. nw 4. The set of counting numbers greater than 20. nw 5. The set of intelligent students.  Let’s try! Determine whether the set is finite or infinite. If it is finite set, give its cardinality. F 1. The set of the first two counting numbers. 2 F 2. {a, b, c, d, ……..x, y, z} 26 I 3. The star in the sky. F 4. The set of even numbers between 1 and 11. 5 F 5. {positive whole number factors of 12} 6  Let’s try! Tell whether the statement is TRUE or FALSE. Given: A = {1,3,5,7,9,11} B = { even nos. less than 14 C = { whole nos. between 8 and 14 D = { prime nos. less than 15} T 1. 6∈B F 3. 5 ∉ D F 2. 1 ∈ C T 4. 4∉ A Universal Set and Subsets Universal Set Example: A = {1, 2, 3, 4, 5} is a set that consists B = {4, 5, 6, 7} of all elements being considered in a problem. Then the universal it is represented by set is: the letter U. U = {1, 2, 3, 4, 5, 6, 7}  SUBSET Set A is said to be a subset of B if every element of A is also an element of B. “A B” read as “ A is a subset of B” Proper subset Improper subset Power set If every element If all the elements Is the set of all the of A is also an element of A is also the the subsets of a of B and B contains elements of B or given set. at least one element simply they are equal which is not in A. sets.  Examples: Find all the subsets of A = {1, 2, 3} Proper subsets: {1}, {2}, {3}, {1,2},{1,3},{2,3}, { } Improper subset: {1, 2, 3} Power set: { { }, {1}, {2}, {3},{1,2},{1,3},{2,3},{1,2,3} } ▪ {1} is a proper subset of A because not all the elements of A are found in {1} and 1 is the element of A. ▪ {1,2,3} is an improper subset of A because all the elements in A are also found in {1,2,3}. ▪ 3 is not a subset of A, because 3 only an element.  Let’s try this! A. Give the universal set of the following set. 1. A = {1, 4, 5}, B = {2, 3, 4}, and C = {5, 6, 7} U = { 1, 2, 3, 4, 5, 6, 7} 2. J = {2, 4}, M = {6, 8, 10} U = { 2, 4, 6, 8, 10}  Let’s try this! B. Find all the subsets of M = { 1, 3, 5, 7}. Proper subsets: { }, {1}, {3},{5}, {7},{1,3},{1,5},{1,7}, {3,5},{3,7},{5,7} Improper subsets: { 1, 3, 5, 7} Power sets: {{ }, {1}, {3},{5}, {7},{1,3},{1,5},{1,7}, {3,5},{3,7},{5,7}, {1,3,5,7} }  Ordered Pair ❑ Given elements a and b, the symbol (a , b) denotes the ordered pair consisting of a and b together with the specification that a is the first element and ` is the second element. Two ordered pairs (a , b) and (c , d) are equal if , and only if, a = c and b = d. (a , b) = (c , d) means that a = c and b = d  Ordered Pair Example: ▪ Is (1 , 3) = (3, 1) ? No, by definition of equality of ordered pairs. 1 2 ▪ Is ( , 6) = ( , 36) ? 4 8 Yes, by definition of equality of ordered pairs. Operation of Sets 1.) Intersection of sets The intersection of two sets A and B denoted by A ∩ B is the set of all elements which are common to both A and B. Example: A B A = {2, 4, 5, 6 } 2 4 3 B = {3, 4, 5, 7} 6 5 7 A ∩ B = { 4, 5} 1.) Intersection of sets Joint sets – are sets with at least one common elements. Example: A B A = {2, 4, 5, 6 } 2 4 3 B = {3, 4, 5, 7} 6 5 7 A ∩ B = { 4, 5}  Disjoint sets – sets with no common elements. Example: A = {1, 3, 5} B = {2, 4, 6} 1, 3, 5 2, 4, 6 A ∩ B = { } or ∅ 2.) Union of sets The union of two sets A and B denoted by A U B is the set of all elements which belong to A or to B or to both A and B. Example: 2 1 A = {2, 4, 5, 6 } 4 5 B = {1,3, 5} 6 3 A U B = { 1, 2, 3, 4, 5, 6} 3.) Complement of a Set The complement of a set A , is written A’, is the set of elements in a universal set that are not in A. Example: U = { 1, 2, 3, 4, 5, 6, ,7 ,8 ,9 ,10} A = { 2, 4, 6, 8,10} A’ = { 1, 3, 5, 7, 9} Other Example: U = { 1, 2, 3, 4, 5} A = {3, 4} C = { 1, 2, 3, 4, 5} B = {2, 3, 5} D={ } A’ = { 1, 2, 5} C’ = { } B’ = { 1, 4} D’ = { 1, 2, 3, 4, 5} 4.) Difference of Two Sets The difference of sets, is the set of all elements of A that are not elements of B, in symbol, this is written A–B. Example: A = {1, 2, 3, 4, 5} A = {m, a, t, h} B = {1, 3, 5} B = {m, a, t} A – B = {2, 4} A – B = {h} 5.) Product of Two Sets Let A and B any two sets, then the product of the two sets A and B is the set of all ordered pairs (x,y) where x is an element of A and y is an element of B. A x B = (x, y) Example: A = {1, 2} B = {d, o, g} A x B = {(1, d), (2,d),(1,o),(2,o),(1,g),(2,g)} Example: A = {a, b, c} B = {2, 4, 6} BxA = {(2, a), (4,a),(6,a),(2,b),(4,b),(6,b),(2,c),(4,c),(6,c)} A = {1,2} B = {x,y} A x B = { (1,x),(2,x),(1,y),(2,y)} Let’s try this! U = { 1,2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1,2, 3, 4} C = { 7, 8, 9, 10} B = { 2, 4} D = { 7, 9} 1. A ∩ B 2. A U B A ∩ B = {2,4} A U B = {1, 2, 3, 4} Let’s try this! U = { 1,2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1,2, 3, 4} C = { 7, 8, 9, 10} B = { 2, 4} D = { 7, 9} 3. A ∩ C A∩C={ } 4. A U C A U B = {1, 2, 3, 4, 7, 8, 9, 10} Let’s try this! U = { 1,2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 2, 3, 4} C = { 7, 8, 9, 10} B = { 2, 4} D = { 7, 9} 5. A − B 6. C − D A = { 1, 2, 3, 4} C = { 7, 8, 9, 10} B = { 2, 4} D = { 7, 9} A − B = { 1, 3} C – D = { 8, 10} Let’s try this! U = { 1,2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1,2, 3, 4} C = { 7, 8, 9, 10} B = { 2, 4} D = { 7, 9} 7. A’ A’= { 5, 6, 7, 8, 9, 10} 8. D’ D’= { 1, 2, 3, 4, 5, 6, 8, 10} Let’s try this! U = { 1,2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1,2, 3, 4} C = { 7, 8, 9, 10} B = { 2, 4} D = { 7, 9} 9. (A∪ B ) ∩ 𝑪′ A U B = {1, 2, 3, 4} C’= {1, 2, 3, 4, 5, 6} (A∪ 𝐁 ) ∩ 𝑪′ = {𝟏, 𝟐, 𝟑, 𝟒} Let’s try this! U = { 1,2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1,2, 3, 4} C = { 7, 8, 9, 10} B = { 2, 4} 10. B x D B = { 2, 4} D = { 7, 9} B x D ={(2,7),(2,9),(4,7),(4,9)}  QUIZ 1 I. Use the roster method or the listing method to specify the elements of the following sets. 1. The counting numbers from 2 to 10. 2. The set of months ending in ber. 3. The set of even integers from 5 to 20. 4. The set of consonant letters from the word MATHEMATICS. 5. The set of odd numbers from 10 to 15. II. Which of the following sets is finite, infinite or empty set? 6. The set of months in a year. 7. The set of even numbers greater than 100. 8. The set of multiples of 5 between 20 and 500. 9. The set of negative integers greater than zero. 10. The set of all lines on a plane passing through a given point. III. Solve for what is asked. U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 3, 5, 7, 9} C = {2, 4, 8} B = {2, 4, 6, 8, 10} D = {3, 6, 9} Find: 11. A’ = 16. A ∪ B = 12. B’ = 17. A ∩ B = 13. D’ = 18. B ∩ C = 14. B–C= 19. C ∪ D = 15. C–D= 20. A ∩ C = Thank you!