# Re: The seven step-Mathematical preliminaries 2

```Hi Marty,
```
```
On 04 Jun 2009, at 01:11, m.a. wrote:

> Bruno,
>            I stopped half-way through because I'm not at all sure of
> my answers and would like to have them confirmed or corrected, if
> necessary, rather than go on giving wrong answers.   marty a.

No problem.

>
>
> Exercise 1: Could you define in intension the following infinite set
> C = {101, 103, 105, ...}
> C = ?                          C={x such that x is odd & x <101}

I guess you meant C = {x such that x is odd and x > 101}.  ">" means
"bigger than", and "<" means little than. OK.

>
> 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}.

Indeed 0 is little than 10, and 0 is a multiple of four (because 0 =
4*0), and
4  is little than 10, and 4 is a multiple of four (because 4 = 4*1)
8 is little than 10, and 8 is a multiple of 4 (because 8 = 4*2)
The next mutiple of 4 is 12. It cannot be in the set because 12 is
bigger than 10.
The numbers 1, 2, 3, 5, 7, 9 cannot be in D, because they are not
multiple of 4. You cannot write 1 = 4 * (some natural numbers), nor
can you write 3 or 5, or 7 or 9 =  4 * x with x a natural number.

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 second group yet it is included in

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".  I guess you try to go to much quickly, or that the font of
your computer are too little, or that you have eyesight problems, or
that you have some dyslexia.

>
> Exercice 3.
> Let N = {0, 1, 2, 3, ...}
> Let A = {x ⎮ x < 10}
> Let B = {x ⎮ x is even}
> Describe in extension (that is: exhaustion or quasi-exhaustion) the
> following sets:
>
> N ∪ A = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}
> N ∪ B = {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}
> A ∪ B = {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}
> B ∪ A = {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}

All that would be correct if you were searching the intersection, but
"∪" is the UNION symbol. (and "∩" is the INTERSECTION symbol).

also you wrote the "⎮" as "inter", instead of "such that".

>
> N ∩ A = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}
> B ∩ A =  {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}
> N ∩ B =  {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}
> A ∩ B =   {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}

All that is correct. Careful you were still using "inter" in place of
"such that". Your last line should be

A ∩ B =   {x such that x <10} inter {x such that x is even}=
{0,2,4,6,8}

>
> Exercice 4
>
> Is it true that A ∩ B = B ∩ A, whatever A and B are?       yes
> Is it true that A ∪ B = B ∪ A, whatever A and B are?      yes

Both are correct.

Not bad Marty!  Just read carefully. I thing you have just dismiss the
paragraph were I define "UNION". And then, you or your computer seems
to have a trouble in distinguishing the symbols "∩" and "∪".

Example
{1, 2, 3} ∩  {3, 4, 5} = {3}
{1, 2, 3} ∪  {3, 4, 5} = {1, 2, 3, 4, 5}

Tell me if it is OK, now.

And then I let you think on the next exercises. Take the time to read
slowly. Have you a problem of dyslexia? Do you see the difference
between "<" and ">" ?
If there is a problem with the symbols, I can switch on "english
symbol".

Have a nice week-end, don't hesitate to ask questions, clarification
of points, or more examples.

Bruno

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

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