You are quick!

On 02 Jul 2009, at 18:42, m.a. wrote:

>
>
> Could you tell me if you understand and/or remember those  
> definitions (where a and b denoting arbitrary sets):
>
> (a INTERSECTION b) = {x SUCH-THAT (x BELONGS-TO a) and (x BELONGS-TO  
> b)}
>
> (a UNION b) = {x SUCH THAT (x BELONGS-TO a) or (x BELONGS-TO b)}
>
> Can you compute
>
> {1, 2, 7, 789} UNION {1, 2, 7, 5678} = ?  1,2,7,789, 5678
Almost OK.
{1, 2, 7, 789} UNION {1, 2, 7, 5678} = {1,2,7,789, 5678}.
Don't forget the accolades, which means that you have as result the  
SET {1,2,7,789, 5678}

> {1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = ? 1, 2, 7, 789

Not correct. To belong to A INTERSECTION B, the element must belong to  
A, *and* must belong to B. 1, 2 and 7 does belong indeed to A and to  
B, in this case, with A = {1, 2, 7, 789}, and B = {1, 2, 7, 5678}),  
but neither 789, nor 5678 do belong to both A and B.
So {1, 2, 7, 789} INTERSECTION {1, 2, 7, 5678} = {1, 2, 7}

Just tell me if you agree.

>
> Do you remember the empty set? Can you compute:
> {1, 2} UNION { } = ?  1,2


OK, but don't forget the accolades.
{1, 2} UNION { } = ?  {1,2}


> {1} UNION { } =  { }
You are too quick here, you forget to type the 1.
{1} UNION { } =  {1 }


> {1, 2, 3} UNION {1, 2, 3} = ? 1,2,3
OK (my mind adds the accolades)

> { } UNION { } = ? { }
Very good. You could eliminate the "?".

> {1, 2} INTERSECTION { } = ? { }

Excellent.


> {1} INTERSECTION { } = ? { }

Bravo.


> {1, 2, 3} INTERSECTION {1, 2, 3} = ? 1, 2 3

Exact. (well, I continue to add the accolades, and eliminate the "?")

> { } INTERSECTION { } = ?  { }

Exact. In this case you see how much it is important to not forget the  
accolades!


>
>
> Now, an important distinction which will follow us through ...  
> forever.  I suggest you read attentively the next two paragraphs two  
> times before breakfast, every day for one week. :), Really take all  
> your time. It concerns the notion of operation, and relation.
>
> INTERSECTION and UNION, are operations on sets, like addition (+, or  
> PLUS) and multiplication (*, or TIMES) are operation on numbers.  
> This means, typically, that, if x and y denote numbers, then x + y,  
> and x * y, will denote, or are equal to, numbers. For example 3 + 4  
> is equal to 7.
> Similarly, if x and y denotes, or are equal, to sets, then x  
> INTERSECTION y denotes, or is equal to, some set. For example {1,2}  
> INTERSECTION {2, 7} is equal to some set, actually the set {2}. OK?
>
> Operations are important, as you can guess, but relations are as  
> well important. Operations lead to new elements, new objects. From  
> the numbers 2 and 3, you get the element 5. Relations pertains or  
> does not pertain, or equivalently, leads to true or false.
>
> Example. The relation LESS-THAN, among the numbers. (x LESS-THAN y)  
> is true if x is less than y. So (3 LESS-THAN 56) is true, and (56  
> LESS-THAN 3) is false. An important relation pertaining on sets is  
> the relation of inclusion, or of being a subset of a set.
>
> By definition a set x will be said included in y (or be said subset  
> of y), when all the elements of x are among the elements of y. We  
> will write (x INCLUDED-IN y) when the set x is included in the set y.
> For example, the set {1, 2} is included in the set {3, 2, 1}, but is  
> not included in the set {3, 1}.
>
> Exercise: in the following, what is true or false?
>
> 45 LESS-THAN 67  true

OK.


> 0 LESS-THAN 1   true

OK.


> 999 LESS-THAN 4  false

OK.

> {1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true

OK.


> {1} INCLUDED-IN {1, 2} true

OK.

>
>
> oops, I must go. You are lucky ;)

I'm back!  I give you two last exercises to ponder about, just  in  
case of insomnia. Again, take your time. I hope Kim follows, and does  
not look at the solution !


1°) In the two relational formula below, one is true, the other is  
false. Which one are what?

a)    { } INCLUDED-IN { }
b)    { } BELONGS-TO { }

2°) And I give you a slightly longer exercise. Can you give me all the  
subsets of the set {1, 2} ?. That is, can you give me all the sets  
which are included in the set {1, 2} ? In case of doubt, reread the  
definitions, reread the examples, and never panic! I give you a hint:  
the set {1, 2} has four subsets. Can you find them?

Good job, Marty.

Bruno

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




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