# Re: The seven step series

From: m.a. <marty684.domain.name.hidden>
Date: Thu, 2 Jul 2009 18:18:48 -0400

Bruno,
Comments and questions are interspersed below.
marty
----- Original Message -----
From: Bruno Marchal
To: everything-list.domain.name.hidden
Sent: Thursday, July 02, 2009 1:44 PM
Subject: Re: The seven step series

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 07 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. I agree and can't understand how I could have been so careless.

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

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

{1} UNION { } = { }
You are too quick here, you forget to type the 1.
{1} UNION { } = {1 } Yes, I mistook the {1} for the number of the question...not part of the equation. I tend to overlook the fine points.

{1, 2, 3} UNION {1, 2, 3} = ? 1,2,3

{ } 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 03 + 04 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?......No!

Why not the sets {1,2,7} if INTERSECTION means BOTH?

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.

00 LESS-THAN 01 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 { }True
b) { } BELONGS-TO { } True

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?

{1} {2} {1,2} {2,1} why not {3} ?

Good job, Marty.

Bruno

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

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Received on Thu Jul 02 2009 - 18:18:48 PDT

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