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Lect 6 - October 14, 2008 - ECS 20 - Fall 2008 - Phil Rogaway
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Today: o Quiz #1  (20 mins)
       o Set theory, continued 

NOTE: Sorry but these notes are rather sketchy ....

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1. Specifying Sets and Set Operations
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\N, \R, \Z, \Q

What set is this?
Many ways to specif sets: 

   P = {n: n is a prime number}
   Let P be the set of prime numbers.
   P = {n \in N :  i | n -> i \in {-n,-1,1,n}} 
   P = {2,3,5,7,11,...}

   
No good:

   No number divides it.
   P = {primes}

Operations
   =     \ne
   \in   \not\in   
   \subset \subseteq   \supset  \supseteq

   union 
   intersection
   complement *alwasy relative to some universe U*   xor
   set difference
   exclusive or     A\B \cup B\A 

   define DISJOINT: Sets A and B are *disjoint* if
                    \neg \exists a\in A\cap B
   define when a set is FINITE
            exists a  function 
   define when a set is INFINITE

  Draw some VENN DIAGRAMS
  A \cup (B\cap C)


"Laws of the algebra of sets:
   A u A = A                         A\cap A = A
   A u (B u C) = (A u B) u C         A\cap (B \cap C) = (A\cap B)\cap C
   A u B = B u A                     A\cap B = B\cap A
   A u (B\cap C) = A u C \cap B u C  A \cap (B u C) = A\cap B u A\cap C
   A u \emptyset = A                 A \cap \emptyset = \emptyset
   A u U         = U                 A \cap U = A
   (A^c)^c       = A 
   A u A^c = U                       A \cap A^c = \emptyset
   U^c = \emptyset                   \emptyset^c = U
   (A u B)^c = A^c \cap B^c        (A\cap B)^c = A^c u B^c     <-- DeMorgan's laws

Prove one, say the first.
  a \in (A\cup B)^c iff
  a \not\in A\cup B
  a \not\in A  and   a \not\in B
  a \in A^c and a \in B^c
  a \in A^c \cap a\in B^c

|S| = the number of element in S if S is finite, \infinity otherwise
A= {{a},b,\emptyset}.   |A| =  3 
A = {emptyset, {emptyset}, {{emptyset}}}     |A|=3.


AVOID TYPE ERRORS!!!
... in a proof, x and y are real numbers.    "Suppose x\subseteq y" .... 
   and many other such problems.


CROSS PRODUCT 

A x B = {(a,b): A\in A, B\in B}

UNORDERED PRODUCT

A & B = {{a,b}: A\in A, B\in B}    // when I learned graph theory -- never saw it since!

\R^2  points in the plane

Do problem 5 from the HW:
  
  The variables are V = {P_{i,j}^k | i,j \in \N, k \in [1..t]}

  { OR_{i,j,k}  P_{i,j}^k | (i,j)\in \N^2} u
  { AND_{k,k}  P_{i,j}^k -> \neg P_{i,j}^k'} u 
  { P_{i,j}^k \land \neg P_{i,j+1}^k' -> S(k,k') | k\ne k\, i,j\in \N} u  
  { P_{i,j}^k \land \neg P_{i+1,j}^k' -> T(k,k') | k\ne k\, i,j\in \N} 





Exercise: carefully write an expression for the following sets:
 - A tile type is a four-tuple of integers \N \times \N \times \N \times N.


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2. Sets STRINGS (elements of formal language theory) 
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Define and give examples:
 Alphabet  
 String          operation: concatenation
 Language        extend concatenation to this set
 Class           a set of languages

 Empty string (\epsilon) (\Lambda)

 A*
 A^0 = {\emptyset}    (why? For recursive definition clean)

 \Sigma^*

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3. Elements of counting 
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If A and B are disjoint finite sets, then
  |A\cup B| = |A| \cup |B|


More generally,
    |A\cup B| = |A| + |B| - |A\cap B|
 for any finite sets A, B
  "inclusion-exclusion principle"


If  ecs20 has  70 students and
    ecs30 has 100 students and
    10 students are taking both classes,
    how many are taking ecs20 or ecs30? 
 
    70 + 100 -  10 = 160