Sets and Set Notation PDF

SETS AND SET NOTATION © 2024 SJCS  SET It is a well-defined collection of distinct items or objects © 2024 SJCS  SET It is a well-defined collection of distinct items or objects It is usually denoted by a...

SETS AND SET NOTATION © 2024 SJCS  SET It is a well-defined collection of distinct items or objects © 2024 SJCS  SET It is a well-defined collection of distinct items or objects It is usually denoted by a capital letter The objects or items are called elements. © 2024 SJCS  SET Here is an example of a set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} © 2024 SJCS  SET Here is an example of a set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} denoted by a capital letter © 2024 SJCS  SET Here is an example of a set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} elements © 2024 SJCS  SET Given set Y = {1, 3, 5,... ,55, 57, 59} 13 Y © 2024 SJCS  SET Given set Y = {1, 3, 5,... ,55, 57, 59} 20 Y © 2024 SJCS  SET Given set Y = {1, 3, 5,... ,55, 57, 59} 62 Y © 2024 SJCS  SET Given set Y = {1, 3, 5,... ,55, 57, 59} 45 Y © 2024 SJCS  TYPES OF SETS Subset of a set A B if every element of A is an element of B © 2024 SJCS  TYPES OF SETS Proper Subset of a set A B if A is a subset of B and B has at least 1 element that is not in A © 2024 SJCS  SET Given sets A and B. A = {2,4,6} B = {1,2,4,6} © 2024 SJCS  SET Given sets A and B. A = {2,4,6} B = {1,2,4,6} A B every element of A is an element of B © 2024 SJCS  SET Given sets A and B. A = {2,4,6} B = {1,2,4,6} A B There is at least one element in B that is not in A © 2024 SJCS  SET EXAMPLE A = {w, x, y, z} Name all the subsets and proper subsets of set A © 2024 SJCS  SET A = {w, x, y, z} No element: { } One element: { w}, {x}, { y}, { z} Two elements: { w, x}, { w, y}, { w, z}, { x, y} , { x, z}, { y, z} Three elements: { w, x, y}, { w, x, z}, { w, y, z}, { x, y, z} Four elements: { w, x, y, z}, Total number of subsets: 16 © 2024 SJCS  TYPES OF SETS Number of Subsets and Proper Subsets 2n n Subsets 2 –1 Proper Subsets Where n is the number of elements in a set © 2024 SJCS  TYPES OF SETS Universal Set Denoted by U, is the set of all possible elements of any set used in the problem. © 2024 SJCS  TYPES OF SETS Empty Set or Null Set It is a set with no elements denoted by { } or An empty set is always a subset of any set © 2024 SJCS  SET EXAMPLES A positive integer less than 1 A set of 10-year old teachers from your school © 2024 SJCS Define the following: SET © 2024 SJCS Define the following: SET SUBSET © 2024 SJCS Define the following: SET SUBSET PROPER SUBSET © 2024 SJCS Define the following: SET SUBSET PROPER SUBSET EMPTY SET © 2024 SJCS Define the following: SET SUBSET PROPER SUBSET EMPTY SET UNIVERSAL SET © 2024 SJCS Define the following: SET SUBSET PROPER SUBSET EMPTY SET UNIVERSAL SET CARDINALITY OF A SET © 2024 SJCS  SET PRACTICE EXERCISE A = {1,2,3,...,43,44,45} B = {x| x is a multiple of 3, 1 < x < 30} C = {even number from 1 to 18} 12 C © 2024 SJCS  SET PRACTICE EXERCISE A = {1,2,3,...,43,44,45} B = {x| x is a multiple of 3, 1 < x < 30} C = {even number from 1 to 18} 32 B © 2024 SJCS  SET PRACTICE EXERCISE A = {1,2,3,...,43,44,45} B = {x| x is a multiple of 3, 1 < x < 30} C = {even number from 1 to 18} 16 C © 2024 SJCS  SET PRACTICE EXERCISE A = {1,2,3,...,43,44,45} B = {x| x is a multiple of 3, 1 < x < 30} C = {even number from 1 to 18} 17 A © 2024 SJCS  SET PRACTICE EXERCISE A = {1,2,3,...,43,44,45} B = {x| x is a multiple of 3, 1 < x < 30} C = {even number from 1 to 18} B A © 2024 SJCS  SET PRACTICE EXERCISE A = {1,2,3,...,43,44,45} B = {x| x is a multiple of 3, 1 < x < 30} C = {even number from 1 to 18} C A © 2024 SJCS Operations On sets ∪ Operations On sets  Operations On sets INTERSECTION OF SETS UNION OF SETS Set difference Complement of a set  Operations On sets The intersection of sets A and B is the set of elements that are members of BOTH A and B. INTERSECTION OF SETS The word AND indicates intersection It is written as A∩ B  Operations On sets Given the following sets: A = {a, b, c, d, e} EXAMPLE B = {e, g, k, a} C = {a, g, d, p, q} INTERSECTION OF SETS D = {d, p, b} Find B∩C {a, g}  Operations On sets Given the following sets: A = {a, b, c, d, e} EXAMPLE B = {e, g, k, a} C = {a, g, d, p, q} INTERSECTION OF SETS D = {d, p, b} Find C∩A {a, d}  Operations On sets Given the following sets: A = {a, b, c, d, e} EXAMPLE B = {e, g, k, a} C = {a, g, d, p, q} INTERSECTION OF SETS D = {d, p, b} Find D∩C {d, p}  Operations On sets Given the following sets: A = {a, b, c, d, e} EXAMPLE B = {e, g, k, a} C = {a, g, d, p, q} INTERSECTION OF SETS D = {d, p, b} Find (A ∩ C) ∩ B {a}  Operations On sets Given the following sets: A = {a, b, c, d, e} EXAMPLE B = {e, g, k, a} C = {a, g, d, p, q} INTERSECTION OF SETS D = {d, p, b} Find D∩B { }  TYPES OF SETS Disjoint Sets Sets with no common elements. Give an example of disjoint sets The intersection of disjoint sets is an empty set.  Operations On sets INTERSECTION OF SETS UNION OF SETS Set difference Complement of a set  Operations On sets The UNION of sets A and B is the set of elements that are in EITHER A, or B, or both. The word OR indicates union. UNION OF SETS It is written as A ∪B  Operations On sets Given the following sets: A = {a, b, c, d, e} B = {e, g, k, a} EXAMPLE C = {a, g, d, p, q} UNION OF SETS D = {d, p, b} Find A∪B {a, b, c, d, e, g, k}  Operations On sets Given the following sets: A = {a, b, c, d, e} B = {e, g, k, a} EXAMPLE C = {a, g, d, p, q} UNION OF SETS D = {d, p, b} Find B∪C {a, d, e, g, k, p, q}  Operations On sets Given the following sets: A = {a, b, c, d, e} B = {e, g, k, a} EXAMPLE C = {a, g, d, p, q} UNION OF SETS D = {d, p, b} Find C∪A {a, b, c, d, e, g, p, q}  Operations On sets Given the following sets: A = {a, b, c, d, e} B = {e, g, k, a} EXAMPLE C = {a, g, d, p, q} UNION OF SETS D = {d, p, b} Find D∪C {a, b, d, g, p, q}  Operations On sets Given the following sets: A = {a, b, c, d, e} B = {e, g, k, a} EXAMPLE C = {a, g, d, p, q} UNION OF SETS D = {d, p, b} Find (A ∪ C) ∪ B {a, b, c, d, e, g, k, p, q}  Operations On sets Given the following sets: A = {a, b, c, d, e} B = {e, g, k, a} EXAMPLE C = {a, g, d, p, q} UNION OF SETS D = {d, p, b} Find D∪B {a, b, d, e, g, k p}  Operations On sets EXERCISES: A = {a| a is a prime number less than 10} B = {b| b is a counting number < 6} C = {c| c is a composite number from 2 to 13}  Operations On sets EXERCISES: A = {a| a is a prime number less than 10} B = {b| b is a counting number < 6} C = {c| c is a composite number from 2 to 13} A B = CB = B C = A B = A C =  Operations On sets EXERCISES: A = {2, 3, 5,7} B = {1, 2, 3, 4, 5} C = {4, 6, 8, 9, 10, 12}  Operations On sets EXERCISES: A = {2, 3, 5, 7} B = {1, 2, 3, 4, 5} C = {4, 6, 8, 9, 10, 12} A B = {2, 3, 5} CB = {4} B C = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12} A  B = {1, 2, 3, 4, 5, 7} A C = { }  Operations On sets EXERCISES: A = {2, 3, 5,7} B = {1, 2, 3, 4, 5} C = {4, 6, 8, 9, 10, 12} ( A  B)  C = (C  B)  A = B  AC =  Operations On sets EXERCISES: A = {2, 3, 5,7} B = {1, 2, 3, 4, 5} C = {4, 6, 8, 9, 10, 12} ( A  B)  C = {2, 3, 5, 4, 6, 8, 9, 10, 12} (C  B)  A = {4, 2, 3, 5,7} B  AC = {1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 12}  Operations On sets INTERSECTION OF SETS UNION OF SETS Set difference Complement of a set  Operations On sets The complement of a set A, is the set of all elements in the universal set (U) that are not in set A. Complement of a set It can be represented as: ‘ A = A =A C  Operations On sets Given the following sets: U = {1, 2, 3, 4, 5} EXAMPLE A = {1, 3, 5} B = {1, 5} Complement of a set C={} D= {4} FIND: A’ {2, 4}  Operations On sets Given the following sets: U = {1, 2, 3, 4, 5} EXAMPLE A = {1, 3, 5} B = {1, 5} Complement of a set C={} D= {4} FIND: C B {2, 3, 4}  Operations On sets Given the following sets: U = {1, 2, 3, 4, 5} EXAMPLE A = {1, 3, 5} B = {1, 5} Complement of a set C={} D= {4} FIND: C {1, 2, 3, 4, 5}  Operations On sets Given the following sets: U = {1, 2, 3, 4, 5} EXAMPLE A = {1, 3, 5} B = {1, 5} Complement of a set C={} D= {4} FIND: C D {1, 2, 3, 5}  Operations On sets EXERCISES: U = {a| a is a composite number less than 17} A = {6, 9, 14} B = {8, 10, 12} C = {4, 9, 12, 16}  Operations On sets EXERCISES: U = {4, 6, 8, 9, 10, 12, 14, 15, 16} A = {6, 9, 14} B = {8, 10, 12} C = {4, 9, 12, 16} 1. A = {4, 8, 10, 12, 15, 16} 4. (A U C)’ = {8, 10, 15} {6, 9, 14, 4, 12, 16} 2. B’ = {4, 6, 9, 14, 15, 16} 5. (B U C)C = {6, 14, 15} {8, 10, 12, 4, 9, 16} 3. CC= {6, 8, 10, 14, 15} 6. (A ∩ C)’ = {4, 6, 8, 10, 12, 14, 15, {9} 16}  Operations On sets INTERSECTION OF SETS UNION OF SETS Set difference Complement of a set  Operations On sets The set difference of sets A and B is the set of all elements in A that are not in B. Set difference It is denoted by A–B  Operations On sets Given the following sets: A = {a, b, c, d} EXAMPLE B = {a, c, f, g} C = {a, f, j, k} Set difference FIND: A–B {b, d}  Operations On sets Given the following sets: A = {a, b, c, d} EXAMPLE B = {a, c, f, g} C = {a, f, j, k} Set difference FIND: B–A {f, g}  Operations On sets Given the following sets: A = {a, b, c, d} EXAMPLE B = {a, c, f, g} C = {a, f, j, k} Set difference FIND: A–C {b, c, d}  Operations On sets Given the following sets: A = {a, b, c, d} EXAMPLE B = {a, c, f, g} C = {a, f, j, k} Set difference FIND: C–A {f, g, k}  Operations On sets Given the following sets: A = {a, b, c, d} EXAMPLE B = {a, c, f, g} C = {a, f, j, k} Set difference FIND: B–C {c, g}  Operations On sets Given the following sets: A = {a, b, c, d} EXAMPLE B = {a, c, f, g} C = {a, f, j, k} Set difference FIND: C–B {j, k}  Operations On sets Practice: U = {a| a is a composite number less than 20} A = {6, 9, 8, 14, 18} B = {6, 8, 10, 12, 18} C = {6, 8, 9, 12, 16} 1. (A-B) = 4. (A U B)’- C = 2. (B-C)’ = 5. B U CC = 3. CC – A = 6. (A ∩ B)C - (B ∩ C)C =  Operations On sets Practice: U = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18 } A = {6, 9, 8, 14, 18} B = {6, 8, 10, 12, 18} C = {6, 8, 9, 12, 16} 1. (A-B) = {4, 6, 8, 10, 12,15, 16, 18 } { 9, 14,} 2. (B-C)’ = {4, 6, 8, 9, 12, 14, 15, 16} {10, 18} 3. CC – A = {4, 10, 15 } {4, 10, 14, 15, 18 }  Operations On sets Practice: U = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18 } A = {6, 9, 8, 14, 18} B = {6, 8, 10, 12, 18} C = {6, 8, 9, 12, 16} 4. (A U B)’- C = {4, 15} {4, 15, 16} 5. B U CC = {4, 6, 8, 10, 12, 14, 15, 18 } C C = {4, 10, 14, 15, 18} 6. (A ∩ B)C - (B ∩ C)C = {12} {4, 9, 10, 12, 14, 15, 16} {4, 9, 10, 14, 15, 16, 18} Boardwork Given: U = {a, b, c, … x, y, z} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: A∪B {a, b, c, d, e, f, g, h} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: B∪D {a, f, g, h, x, b} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: A∩C {a, c} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: B∩D {a, f} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: (B’)C B = {a, f, g, h} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: A∩B∩D {a} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: A C {f, g, h, … x, y, z} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: B C {b, c, d, e, I, j, k, … x, y, z} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: A–C { b, d, e} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: (D – B)’ {x, b} {a, c, d, … y, z} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: A ∪ (D ∩ B) {a,b,c,d,e,f} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: B∪∅ {a, f, g, h} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: B’ ∩ CC {b, d, e, j, k, l, …} Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: (B ∩ C) ∩ AC { } Given: U = {x|x is a letter in the English alphabet} A = {a, b, c, d, e}C = {a, c, h, i} B = {a, f, g, h} D = {x, f, b, a} What is: (A ∩ B) ∪ (A ∩ D) {a , b}