1. Set Operators

2. Copyright © Peter Cappello 20112 Union Let A and 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.

3. Copyright © Peter Cappello 20113 Intersection Let A and 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 and B are disjoint when A  B = .

4. Copyright © Peter Cappello 20114 Difference Let A and B be sets. The difference of A and 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.

5. Copyright © Peter Cappello 20115 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.

6. Copyright © Peter Cappello 20116 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

7. Copyright © Peter Cappello 20117 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

8. 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  false )  x  A A   = A Copyright © Peter Cappello 20118

9. 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.

10. Copyright © Peter Cappello 201110 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  )

11. Copyright © Peter Cappello 201111 Prove A  B = A  B Membership Table AB A  B AB A  BA  B FFFTTTT FTTFTFF TFTFFTF TTTFFFF 1 2 3 4 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.”

12. 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? Copyright © Peter Cappello 201112

13. Copyright © Peter Cappello 201113 Computer Representation of Sets There are many ways to represent sets. Which is best depends on the particular sets & operations. 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. Consider a Java set class, where | U | is a constructor parameter. (In Java, Set is the name of an interface.) –What data structures might be good? –What public methods might you want? –How would you implement them? –Java Set interfaceJava Set interface