On 04 Jul 2009, at 15:17, m.a. wrote:

> Bruno,
>           Can you provide definitions of  "belongs-to"  and   
> "included-in" that distinguish them from "union" and "intersection"?


"Belongs-to" and "included-in" are relations. Their value are true or  
false.


1) (x belongs-to A) means that the object x belongs to the set A.

Examples: 3 belongs-to {3, 4}, because 3 is an element of the set {3,  
4}, or, put in another way (which can be useful for later):

  (3 belongs-to {3, 4}) = true


2) Similarly (x included-in y) is a relation bearing on sets, and (x  
included-in y) = true, means that x is included-in y, and this means  
that all elements of x are elements of y. You don't need to know more,  
but if you want you can define (x included-in y) by

For any z ((z belongs-to x) -> (z belongs-to y)). But I intended to  
introduce "->" later, so don't worry.

Example {3, 4} is included in {3,5,4} because all elements of {3, 4}  
are in {3, 5, 4}. We can write ({3,4} included-in {3,5,4}) = true.  
Another example is ({3, 4} included-in {3,6,7,9,567}) is false,  
because not all elements of {3,4} are in {3,6,7,9,567} (indeed 4  
belongs to {3,4} and not to  {3,6,7,9,567}.

Union and intersection are not relation, but operation.

({3,4} union {4,5}) is not equal to true or to false, it is equal to a  
set: actually the set you get by the union of {3,4} with {4,5}, and  
this is {3,4,5}.
Likewise, the intersection of {3,4} with {4,5}, that is ({3,4} union  
{4,5}) is not true or false, but is equal to {4}.

So:

({3,4} included-in {3,4,5}) = true
({3,4} union {3,4,5}) = {3,4,5}

You see the difference. It is the difference between "the brother of  
Paul", which denotes a human. and "Paul is greater than Julia", which  
is true or false.

Or, on the numbers, less-than (<)  is a relation, and addition and  
multiplication are operation:

(3 <  7) = true
(3+4) = 7

(7 <  3) = false
(7*3) = 21

({ } included-in {3,4}) = true
({1,2} intersection {2,7}) = {2}.

Does this help?

> Here we met a set of sets.
> The set of subsets of a set, can only be, of course, a set of sets.  
> The set {2, 21, 14} is a set of numbers. The set { { }, {4, 78,  
> 56} } is a set of sets. It has two elements: the empty set {}, and  
> the set of numbers {4, 78, 56}. Do not confuse a number, like 24,  
> and a set, like {24}, which is a set having a number has elements.  
> In particular it is the case that  {4, 78, 56} belongs to { { }, {4,  
> 78, 56} }. Take it easy, and meditate on the following exercise:
>
> Which of the following are true
>
> {3, 5} included-in {3, 5} True

This is correct.


> {3, 5} belongs-to {3, 5}
> {3, 5} included-in { {3, 5} }
> {3, 5} belongs-to { {3, 5} }
>
> Take your time,
>
> Bruno
>
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
>
> >

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




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