# Re: The seven step series

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

Do you remember the empty set? Can you compute:

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

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
0 LESS-THAN 1   true
999 LESS-THAN 4  false
{1, 2, 3} INCLUDED-IN {4, 1, 5, 2, 3, 8} true
{1} INCLUDED-IN {1, 2} true

oops, I must go. You are lucky ;)

Bruno

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

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