Bruno,
              Comments and questions are interspersed below.   
                                                                                
                      marty 
  ----- Original Message ----- 
  From: Bruno Marchal 
  To: everything-list@googlegroups.com 
  Sent: Thursday, July 02, 2009 1:44 PM
  Subject: Re: The seven step series


  You are quick!




  On 02 Jul 2009, at 18:42, m.a. wrote:




      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
  Almost OK. 
  {1, 2, 7, 789} UNION {1, 2, 7, 5678} = {1,2,7,789, 5678}. 
  Don't forget the accolades, which means that you have as result the SET 
{1,2,7,789, 5678}


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


  Not correct. To belong to A INTERSECTION B, the element must belong to A, 
*and* must belong to B. 1, 2 and 7 does belong indeed to A and to B, in this 
case, with A = {1, 2, 7, 789}, and B = {1, 2, 7, 5678}), but neither 789, nor 
5678 do belong to both A and B.
  So {1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = {1, 2, 7}


  Just tell me if you agree.     I agree and can't understand how I could have 
been so careless.




      Do you remember the empty set? Can you compute:
      {1, 2} UNION { } = ?  1,2




  OK, but don't forget the accolades.    Are accolades brackets?
  {1, 2} UNION { } = ?  {1,2}




      {1} UNION { } =  { }
  You are too quick here, you forget to type the 1. 
  {1} UNION { } =  {1 }             Yes, I mistook the {1} for the number of 
the question...not part of the equation. I tend to overlook the fine points.




      {1, 2, 3} UNION {1, 2, 3} = ? 1,2,3
  OK (my mind adds the accolades)


      { } UNION { } = ? { }
  Very good. You could eliminate the "?".


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


  Excellent.




      {1} INTERSECTION { } = ? { }


  Bravo.




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


  Exact. (well, I continue to add the accolades, and eliminate the "?")


      { } INTERSECTION { } = ?  { }


  Exact. In this case you see how much it is important to not forget the 
accolades!








      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?......No!

                                                                                
                                                                                
                                                                Why not the 
sets {1,2,7} if INTERSECTION means BOTH?


      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


  OK.




      0 LESS-THAN 1   true


  OK.




      999 LESS-THAN 4  false


  OK.


      {1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true


  OK.




      {1} INCLUDED-IN {1, 2} true


  OK.






      oops, I must go. You are lucky ;) 


  I'm back!  I give you two last exercises to ponder about, just  in case of 
insomnia. Again, take your time. I hope Kim follows, and does not look at the 
solution ! 




  1°) In the two relational formula below, one is true, the other is false. 
Which one are what?


  a)    { } INCLUDED-IN { }True
  b)    { } BELONGS-TO { } True


  2°) And I give you a slightly longer exercise. Can you give me all the 
subsets of the set {1, 2} ?. That is, can you give me all the sets which are 
included in the set {1, 2} ? In case of doubt, reread the definitions, reread 
the examples, and never panic! I give you a hint: the set {1, 2} has four 
subsets. Can you find them?

                                                                           {1} 
{2} {1,2} {2,1}     why not {3} ?


  Good job, Marty. 


  Bruno


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







  

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