Set Operators Goals Show how set identities are established Introduce some important identities.

Copyright © Peter Cappello2 Union Let A & B be sets. A union B, denoted A  B, is the set A  B = { x | x  A  x  B }. Draw a Venn diagram to visualize this. Example O = { x  N | x is odd }. S = { s  N |  x  N s = x 2 }. Describe O  S.

Copyright © Peter Cappello3 Intersection Let A & B be sets. A intersection B, denoted A  B, is the set A  B = { x | x  A  x  B }. Draw a Venn diagram to visualize this. Example O = { x  N | x is odd }. S = { s  N |  x  N s = x 2 }. Describe O  S. A & B are disjoint when A  B = .

Copyright © Peter Cappello4 Difference Let A & B be sets. The difference of A & B, denoted A – B, is A – B = { x | x  A  x  B }. Draw a Venn diagram to visualize this. Example O = { x  N | x is odd }. S = { s  N |  x s = x 2 }. Describe O – S.

Copyright © Peter Cappello5 Complement Let A be a set. The complement of A is { x | x  A } = U – A. Draw a Venn diagram to visualize this. Example O = { x  N | x is odd}. Describe the complement of O. Since I cannot overline in Powerpoint, I denote the complement of A as A.

Copyright © Peter Cappello6 Set Identities IdentityName of laws A   = A A  U = A Identity A  U = U A   =  Domination A  A = A A  A = A Idempotent Complement of A = AComplementation A  B = B  A A  B = B  A Commutative

Copyright © Peter Cappello7 IdentityName of laws A  (B  C)= (A  B)  C A  (B  C)= (A  B)  C Associative A  (B  C) = (A  B)  (A  C) A  (B  C) = (A  B)  (A  C) Distributive A  B = A  B A  B = A  B De Morgan A  (A  B) = A A  (A  B) = A Absorption A  A = U A  A =  Complement

Think like a mathematician How much is new here? LogicSet x  S S False  TrueUniverse    complement  = Can you mechanically produce set identities from propositional identities via this translation? Example: ( x  A  x   )  x  A A   = A Copyright © Peter Cappello8

9 Prove A  B = A  B Venn diagrams 1.Draw the Venn diagram of the LHS. 2.Draw the Venn diagram of the RHS. 3.Explain that the regions match.

Copyright © Peter Cappello10 Prove A  B = A  B Use set operator definitions 1.A  B = { x | x  A  B } (defn. of complement) 2. = { x |  (x  A  B) } (defn. of  ) 3. = { x |  (x  A  x  B) } (defn. of  ) 4. = { x | (x  A  x  B) } (Propositional De Morgan) 5. = { x | (x  A  x  B) } (defn. of complement ) 6. = A  B (defn. of  )

Copyright © Peter Cappello11 Prove A  B = A  B Membership Table AB A  B AB A  BA  B FFFTTTT FTTFTFF TFTFFTF TTTFFFF AB Let x be an arbitrary member of the Universe. In the table below, each column denotes the proposition function “x is a member of this set.”

Think like a mathematician Is membership table the analog of truth table? With 3 propositional variables, a truth table has 2 3 rows. With 3 sets, do we have 2 3 regions? Does this generalize to n sets? What is the analog of modus ponens? 1.What is the set analog of p  q? 2.What is the set analog of a tautology? If interested, see chapter 12 of textbook. Copyright © Peter Cappello12

Analogy between logic & sets In logic: p  q ≡  p  q Its set analog is P  Q Set analog of modus ponens ( p  ( p  q ) )  q is complement( P  ( P  Q ) )  Q Copyright © Peter Cappello13

Copyright © Peter Cappello14 Computer Representation of Sets There are many ways to represent sets. Which is best depends on the particular sets & operations. Bit string: Let | U | = n, where n is not “too” large: U = { a 1, …, a n }. Represent set A as an n-bit string. If ( a i  A ) bit i = 1; else bit i = 0. Operations , , _ are performed bitwise. In Java, Set is the name of an interface.Set is the name of an interface Consider a Java set class (e.g., BitStringSet), where | U | is a constructor parameter. –What data structures might be useful to implement the interface? –What public methods might you want? –How would you implement them?