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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 specify 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 (Book, 1.4)
   =     \ne
   \in   \not\in   
   \subset \subseteq   \supset  \supseteq

   union 
   intersection
   complement *always 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:(section 1.5) MOre below than done in class.
   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


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3. Elements of counting (Section 1.6)
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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