Could you tell me if you understand and/or remember those definitions (where 
a and b denoting arbitrary sets):


  (a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO b)}


  (a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}


  Can you compute


  {1, 2, 7, 789} UNION {1, 2, 7, 5678} = ?  1,2,7,789, 5678
  {1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789
   

  Do you remember the empty set? Can you compute:


  {1, 2} UNION { } = ?  1,2
  {1} UNION { } = ? { }
  {1, 2, 3} UNION {1, 2, 3} = ? 1,2,3
  { } UNION { } = ? { }
  {1, 2} INTERSECTION { } = ? { }
  {1} INTERSECTION { } = ? { }
  {1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3
  { } INTERSECTION { } = ?  { }




  Now, an important distinction which will follow us through ... forever.  I 
suggest you read attentively the next two paragraphs two times before 
breakfast, every day for one week. :), Really take all your time. It concerns 
the notion of operation, and relation.


  INTERSECTION and UNION, are operations on sets, like addition (+, or PLUS) 
and multiplication (*, or TIMES) are operation on numbers. This means, 
typically, that, if x and y denote numbers, then x + y, and x * y, will denote, 
or are equal to, numbers. For example 3 + 4 is equal to 7.
  Similarly, if x and y denotes, or are equal, to sets, then x INTERSECTION y 
denotes, or is equal to, some set. For example {1,2} INTERSECTION {2, 7} is 
equal to some set, actually the set {2}. OK?


  Operations are important, as you can guess, but relations are as well 
important. Operations lead to new elements, new objects. From the numbers 2 and 
3, you get the element 5. Relations pertains or does not pertain, or 
equivalently, leads to true or false. 


  Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y) is true 
if x is less than y. So (3 LESS-THAN 56) is true, and (56 LESS-THAN 3) is 
false. An important relation pertaining on sets is the relation of inclusion, 
or of being a subset of a set.


  By definition a set x will be said included in y (or be said subset of y), 
when all the elements of x are among the elements of y. We will write (x 
INCLUDED-IN y) when the set x is included in the set y.
  For example, the set {1, 2} is included in the set {3, 2, 1}, but is not 
included in the set {3, 1}.


  Exercise: in the following, what is true or false?


  45 LESS-THAN 67  true
  0 LESS-THAN 1   true
  999 LESS-THAN 4  false
  {1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true
  {1} INCLUDED-IN {1, 2} true




  oops, I must go. You are lucky ;) 


  Bruno










  http://iridia.ulb.ac.be/~marchal/







  

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