# Re: The seven step-Mathematical preliminaries 2

```Bruno,
I don't have dyslexia but my keyboard doesn't contain either the
UNION symbol or the INTERSECTION symbol (unless I want to go into an INSERT
pull down menu every time I use those symbols). I don't need you to switch to
English symbols, but I would like to see the English equivalents of the symbols
you use (so that I can use them). I would also like a reference table defining
each term in both your symbols and their English equivalents which I could look
back to when I get confused. Please include examples. I tend to be somewhat
careless when dealing with very fine distinctions and may type the wrong symbol
while intending to type the correct one. Also, I must admit that the lessons
are going too fast for me and are moving ahead before I've mastered the
previous material. If I'm requesting too much simplification, please let me
know because I'm quite well adjusted to my math disabilities and won't take
offence at all. Thanks,      marty a.```
```

----- Original Message -----
From: Bruno Marchal
Sent: Thursday, June 04, 2009 12:04 PM
Subject: 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

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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