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/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---