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

Copyright © Peter Cappello 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 = x2 }. Describe O  S. Copyright © Peter Cappello

Copyright © Peter Cappello 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 = x2 }. Describe O  S. A & B are disjoint when A  B = . Copyright © Peter Cappello

Copyright © Peter Cappello 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 = x2 }. Describe O – S. Copyright © Peter Cappello

Copyright © Peter Cappello 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 Cappello

Copyright © Peter Cappello Set Identities Identity Name of laws A   = A A  U = A A  U = U A   =  Domination A  A = A A  A = A Idempotent Complement of A = A Complementation A  B = B  A A  B = B  A Commutative Copyright © Peter Cappello

Copyright © Peter Cappello Identity Name 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 Copyright © Peter Cappello

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

Prove A  B = A  B Venn diagrams Draw the Venn diagram of the LHS. Draw the Venn diagram of the RHS. Explain that the regions match. Copyright © Peter Cappello

Prove A  B = A  B Use set operator definitions A  B = { x | x A  B } (defn. of complement) = { x | (x  A  B) } (defn. of  ) = { x | (x  A  x B) } (defn. of  ) = { x | (x  A  x  B) } (Propositional De Morgan) = { x | (x  A  x  B) } (defn. of complement ) = A  B (defn. of  ) Copyright © Peter Cappello

Prove A  B = A  B Membership Table 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.” A B A  B A  B F T A B 3 4 2 1 Copyright © Peter Cappello

Copyright © Peter Cappello Think like a mathematician Is membership table the analog of truth table? With 3 propositional variables, a truth table has 23 rows. With 3 sets, do we have 23 regions? Does this generalize to n sets? What is the analog of modus ponens? What is the set analog of p  q? What is the set analog of a tautology? If interested, see chapter 12 of textbook. Copyright © Peter Cappello

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 Cappello

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 = { a1, …, an }. Represent set A as an n-bit string. If ( ai  A ) bit i = 1; else bit i = 0. Operations ,  , _ are performed bitwise. In Java, 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? Copyright © Peter Cappello