# Re: The seven step-Mathematical preliminaries

```Bruno,
questions.    marty a.
----- Original Message -----
From: "Bruno Marchal" <marc...@ulb.ac.be>```
```

>
>
> ============================================= begin
> ===============================
>
> 1) SET
>
> Informal definition: a set is a collection of object, called elements,
> with the idea that it, the collection or set, can be considered itself
> as an object. It is a many seen as a one, if you want. If the set is
> not to big, we can describe it exhaustively by listing the elements,
> if the set is bigger, we can describe it by some other way. Usually we
> use accolades "{", followed by the elements, separated by commas, and
> then "}", in the exhaustive description of a set.
>
> Example/exercise:
>
> 1) The set of odd natural numbers which are little than 10. This is a
> well defined, and not to big set, so we can describe it exhaustively by
> {1, 3, 5, 7, 9}. In this case we say that 7 belongs to  {1, 3, 5, 7, 9}.
> Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}?    NO
>
> 2) the set of even natural number  which are little than 13. It is {0,
> 2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
> not related to the notion of set, for example they can ask themselves
> if zero (0) is really an even number. We will come back to this.
>
> 3) The set of odd natural numbers which are little than 100. This set
> is already too big to describe exhaustively. We will freely describe
> such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95, 97,
> 99}.
> Exercise 2: does the number 93 belongs to the set of odd natural
> numbers which are little than 100, that is: does 93 belongs to {1, 3,
> 5, 7, 9, 11, ... 95, 97, 99}?
>               YES
>
> 4) The set of all natural numbers. This set is hard to define, yet I
> hope you agree we can describe it by the infinite quasi exhaustion by
> {0, 1, 2, 3, ...}.
> Exercise 3: does the number 666 belongs to the set of natural numbers,
> that is does 666 belongs to {0, 1, 2, 3, ...}.
>          YES
> Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
> 3, ...}?
>                         NO (a guess)
>
>
> 5) When a set is too big or cumbersome, mathematician like to give
> them a name. They will usually say: let S be the set {14, 345, 78}.
> Then we can say that 14 belongs to S, for example.
> Exercise 5: does 345 belongs to S?
>         YES
>
> A set is entirely defined by its elements. Put in another way, we will
> say that two sets are equal if they have the same elements.
> Exercise 6. Let S be the set {0, 1, 45} and let M be the set described
> by {45, 0, 1}. Is it true or false that S is equal to M?
>         YES
> Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
> true or false that S is equal to M?
>              NO
>
> Seven exercises are enough. Are you ready to answer them. I hope you
> don't find them too much easy, because I intend to proceed in a way
> such that all exercise will be as easy, despite we will climb toward
> very much deeper notion. Feel free to ask question, comments, etc. I
> will try to adapt myself.
>                   SO FAR SO GOOD
>
> Next: we will see some operation on sets (union, intersection), and
> the notion of subset. If all this work, I will build a latex document,
> and make it the standard reference for the seventh step for the non
> mathematician, or for the beginners in mathematics.
>
> Bruno
>
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
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