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Module #3 - Sets Sections 2.1 & 2.2: Sets & Set Operations 10/6/2009 (c)2001-2003, Michael P. Frank 1 Module #3 - Sets Introduction • A set is an unordered collection of objects (elements, members) of the set . 10/6/2009 (c)2001-2003, Michael P. Frank 2 Module #3 - Sets Basic notations for sets • (1) listing all of its elements in curly braces: Examples: Finite sets V= {a, e, i ,o ,u } S= {1, 2, 3,…,99} infinite sets: N = {0, 1, 2, …} The Natural numbers. Z = {…, -2, -1, 0, 1, 2, …} The set of integers. Z+ = {1,2,3, …} The set of positive integers. R = The set of “Real” numbers 10/6/2009 (c)2001-2003, Michael P. Frank 3 Module #3 - Sets Basic notations for sets • (2) Set builder notation: For any proposition P(x) over any universe of discourse, {x|P(x)} is the set of all x such that P(x). Example “Set of all odd positive integers less than 10” O= {x | x is an odd positive integer less than 10} 10/6/2009 (c)2001-2003, Michael P. Frank 4 Module #3 - Sets Basic notations for sets • (3) Venn Diagrams U: universal set John Venn 1834-1923 10/6/2009 (c)2001-2003, Michael P. Frank 5 Module #3 - Sets Definition of Set Equality • Two sets are declared to be equal if and only if they contain exactly the same elements. • Example:( order and repition do not matter) {1 ,3 , 5} = {3, 5, 1} = {1, 3, 3, 3, 5, 5, 5} 10/6/2009 (c)2001-2003, Michael P. Frank 6 Module #3 - Sets Membership in sets • x∈S : x is an ∈lement or member of set S. – e.g. 3∈N, “a”∈{x | x is a letter of the alphabet} • x∉S :≡ ¬(x∈S) “x is not in S” – Set equality can be defined in terms of ∈ relation: S=T ↔ (∀x: x∈S ↔ x∈T) 10/6/2009 (c)2001-2003, Michael P. Frank 7 Module #3 - Sets The Empty Set • ∅ (“null”, “the empty set”) is the unique set that contains no elements. • ∅ = {} • Empty set ∅ does not equal the singleton set {∅} ∅ ≠ {∅} 10/6/2009 (c)2001-2003, Michael P. Frank 8 Module #3 - Sets Subset Relation • S⊆T (“S is a subset of T”) means that every element of S is also an element of T. • S⊆T ⇔ ∀x (x∈S → x∈T) • ∅⊆S, S⊆S. If ( S ⊆ T) is true and ( T ⊆ S) is true then S = T ∀x( x∈S ↔ x∈T) • S ⊆ T means ¬(S⊆T), i.e. ∃x(x∈S ∧ x∉T) / 10/6/2009 (c)2001-2003, Michael P. Frank 9 Module #3 - Sets Proper Subset • S⊂T (“S is a proper subset of T”) means every element of S is also an element of T, but S ≠ T. Example: {1,2} ⊂{1,2,3} S T Venn Diagram equivalent of S⊂T 10/6/2009 (c)2001-2003, Michael P. Frank 10 Module #3 - Sets Sets Are Objects, Too! • The objects that are elements of a set may themselves be sets. • E.g. let S={x | x ⊆ {1,2,3}} then S={∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} • Note that 1 ≠ {1} ≠ {{1}} !!!! 10/6/2009 (c)2001-2003, Michael P. Frank 11 Module #3 - Sets Cardinality and Finiteness • |S| (read “the cardinality of S”) is a measure of how many different elements S has. • E.g. |∅|=0 |{1,2,3}| = 3 |{a,b}| = 2 |{{1,2,3},{4,5}}| = 2 10/6/2009 (c)2001-2003, Michael P. Frank 12 Module #3 - Sets The Power Set Operation • The power set P(S) of a set S is the set of all subsets of S. • E.g. P( {a, b}) = {∅, {a}, {b}, {a, b}}. P( {a}) = {∅,{a}) P({∅})= {∅,{∅}} P(∅) = {∅} Note: if a set has n elements then P(S) has 2n elements. 10/6/2009 (c)2001-2003, Michael P. Frank 13 Module #3 - Sets Ordered n- tuples • These are like sets, except that duplicates matter, and the order makes a difference. • Note that (1, 2) ≠ (2, 1) ≠ (2, 1, 1). • Note: 2- tuples are called ordered pairs 10/6/2009 (c)2001-2003, Michael P. Frank 14 Module #3 - Sets Cartesian Products of Sets • For sets A, B, their Cartesian product A×B :≡ {(a, b) | a∈A ∧ b∈B }. • E.g. {a,b}×{1,2} = {(a,1),(a,2),(b,1),(b,2)} • Note that for finite A, B, |A×B|=|A||B|. • Note that the Cartesian product is not commutative: A×B≠B×A. • E.g. {1,2}×{a,b} = {(1,a),(1,b),(2,a),(2,b)} 10/6/2009 (c)2001-2003, Michael P. Frank 15 Module #3 - Sets Section 1.7:Set Operations The Union Operator ∪ • For sets A, B, the A∪B is the set containing all elements that are either in A, or in B or in both. • Formally: A∪B = {x | x∈A ∨ x∈B}. 10/6/2009 (c)2001-2003, Michael P. Frank 16 Module #3 - Sets Union Example {2,3,5}∪{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7} 10/6/2009 (c)2001-2003, Michael P. Frank 17 Module #3 - Sets The Intersection Operator ∩ • For sets A, B, their intersection A∩B is the set containing all elements that are in both A and in B. • Formally: A∩B={x | x∈A ∧ x∈B}. 10/6/2009 (c)2001-2003, Michael P. Frank 18 Module #3 - Sets Intersection Examples • {a,b,c}∩{2,3} = ∅ disjoint • {2,4,6}∩{3,4,5} = {4} 10/6/2009 (c)2001-2003, Michael P. Frank 19 Module #3 - Sets Inclusion-Exclusion Principle • How many elements are in A∪B? |A∪B| = |A| + |B| − |A∩B| • Example: {1,2,3} ∪ {2,3,4,5} = {1,2,3,4,5} {1,2,3} ∩ {2,3,4,5} = {2,3} | {1,2,3,4,5}| = 3+4-2=5 10/6/2009 (c)2001-2003, Michael P. Frank 20 Module #3 - Sets Set Difference • For sets A, B, the difference of A and B, written A−B, is the set of all elements that are in A but not in B. A − B :≡ {x | x∈A ∧ x∉B} = {x | ¬(x∈A → x∈B) } called: complement of B with respect to A. Ex: {1,2,3,4,5,6}-{2,3,5,7,9,11}={1,4,6 } 10/6/2009 (c)2001-2003, Michael P. Frank 21 Module #3 - Sets Set Difference - Venn Diagram Set A−B Set A Set B 10/6/2009 (c)2001-2003, Michael P. Frank 22 Module #3 - Sets Set Complements • U: universe of discourse • A : For any set A⊆U, the complement of A, i.e., it is U−A. • A = {x | x ∉ A} A A • E.g., If U=N, U {3,5} ={0,1,2,4,6,7,...} 10/6/2009 (c)2001-2003, Michael P. Frank 23 Module #3 - Sets Set Identities • Identity: A∪∅ = A = A∩U • Domination: A∪U = U , A∩∅ = ∅ • Idempotent: A∪A = A = A∩A • Double complement: ( A ) = A • Commutative: A∪B = B∪A , A∩B = B∩A • Associative: A∪(B∪C)=(A∪B)∪C , A∩(B∩C)=(A∩B)∩C 10/6/2009 (c)2001-2003, Michael P. Frank 24 Module #3 - Sets DeMorgan’s Law for Sets A∪ B = A ∩ B A∩ B = A ∪ B 10/6/2009 (c)2001-2003, Michael P. Frank 25 Module #3 - Sets Proving Set Identities To prove statements about sets, of the form E1 = E2 (where the Es are set expressions), here are three useful techniques: 1. Prove E1 ⊆ E2 and E2 ⊆ E1 separately. 2. Use set builder notation & logical equivalences. 3. Using set identities 10/6/2009 (c)2001-2003, Michael P. Frank 26 Module #3 - Sets Example: Show A∩B= A∪ B method1: Prove E1 ⊆ E2 and E2 ⊆ E1 • Part 1: Show A∩B ⊆ A ∪ B • Assume x∈ A∩B then : – x ∉ A∩B by the definition of the complement – ~((x ∈ A) ∧ (x ∈ B)) by the definition of intersection – ~(x ∈A) ∨ ~ ( x ∈B) by DeMorgan’s law – x∉A∨ x∉B by the definition of negation – x ∈ A ∨ x ∈ B by the definition of the complement – x∈A∪B by the definition of union This shows that : A∩B ⊆ A ∪ B • Part 2: Show A ∪ B ⊆ A ∩B 10/6/2009 (c)2001-2003, Michael P. Frank 27 Module #3 - Sets Method2: set builder notation • Show A∩B = A ∪ B • A∩B= {x | x ∉ A∩B} = {x | ~( x ∈ (A∩B))} = {x | ~( x ∈ A ∧ x ∈B)} = {x | x ∉ A ∨ x ∉ B} = {x | x ∈ A ∨ x ∈ B)} = {x | x ∈ A ∪ B } = A ∪ B 10/6/2009 (c)2001-2003, Michael P. Frank 28 A ∪ (B ∩ C) Module #3 - Sets Method3 : using set identities • Ex: prove that using set identities: • A ∪ (B ∩ C) = ( C ∪ B) ∩ A 10/6/2009 (c)2001-2003, Michael P. Frank 29

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