On this date, you made the following correction:  "You cannot write D = 4*x 
..., "     But you wrote   D= 4*x   in the exercise just above it. I don't get 
the distinction between your use of the equation and mine.
  ----- Original Message ----- 
  From: Bruno Marchal 

      Exercise 2: I will say that a natural number is a multiple of 4 if it can 
be written as 4*y, for some y. For example 0 is a multiple of 4, (0 = 4*0), but 
also 28, 400, 404, ...  Could you define in extension the following set D = {x 
⎮ x < 10  &  x is a multiple of 4}.    D=4*x  where x = 0 (but also 1,2,3...10)

  You cannot write D = 4*x ..., given that D is a set, and 4*x is a (unknown) 
number (a multiple of four when x is a natural number).

  Read carefully the problem. I gave the set in intension, and the exercise 
consisted in writing the set in extension. Let us translate in english the 
definition of the set D = {x ⎮ x < 10  &  x is a multiple of 4}: it means that 
D is the set of numbers, x, such that x is little than 10, and x is a multiple 
of four. So D = {0, 4, 8}.


  Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 
...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...
  The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 

      A ∩ B = {x ⎮ x ∈ A and x ∈ B}.

      Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}

      Similarly, we can directly define the union of two sets A and B, written 
A ∪ B in the following way:

      A ∪ B = {x ⎮ x ∈ A or x ∈ B}.    Here we use the usual logical "or". p or 
q is suppose to be true if p is true or q is true (or both are true). It is not 
the exclusive "or".

      Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}.   Question: 
In the example above, 5,6 were the intersection because they were the (only) 
two numbers BOTH groups had in common. But in this example, 7 is only in the 
second group yet it is included in the answer. Please explain.

  In the example "above" (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}) we 
were taking the INTERSECTION of the two sets.
  But after that, may be too quickly (and I should have made a title perhaps) I 
was introducing the UNION of the two sets.

  If you read carefully the definition in intension, you should see that the 
intersection of A and B is defined with an "and". The definition of union is 
defined with a "or". Do you see that? It is just above in the quote.

  I hope that your computer can distinguish A ∩ B  (A intersection B) and A ∪ B 
 (A union B).
  In the union of two sets, you put all the elements of the two sets together. 
In the intersection of two sets, you take only those elements which belongs to 
the two sets.

  It seems you have not seen the difference between "intersection" and "union". 

  This has indeed been the case. My usual math disabilities have been 
exacerbated by the confusion of symbols due to E-mail limitations. The 
profusion of little rectangles replacing the UNION symbol make the formulae 
difficult to follow. 


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