# Re: The seven step-Mathematical preliminaries 2

```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}.

SEE BELOW

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,
...}
Etc.

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

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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