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"?

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"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/ --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---