Date: Mon, 22 Oct 2001 03:45:52 -0500 (CDT) From: Mark Jacobson To: Raymond Stantz Subject: Re: Sets & HW 6 > Question 1: One of the things not explained in lecture (or something > that I did not understand) or covered in the text (that I > could tell), was: > > { {n} } what exactly does this mean? Is it the set that contains the > set n? And does this mean that a set can be expressed by containing > numerous subsets? { {n} } <--------- What is between the { and the }? Looks to me like the set contains the set consisting of n, i.e. it contains {n}. { {n} } > For A = { 1, 2, {2} } > > what is the difference between > { {1} } is a proper subset of A <----- FALSE > and {1} is a proper subset of A ? <----- TRUE A subset of B means for all x, if x elementOf A then x elementOf B {1} is a subset of A as shown above since 1 is an elementOf A { {1} } is NOT a subset of A, because {1} is NOT an elementOf A. {1} is an elementOf PowerSet(A), since {1} is a subset of A, and the PowerSet of A is the set of all of its subsets! --- The PowerSet(A) would have 8 elements, its the set of ALL possible subsets of set A. ----------------------- ------- > For A= {1, 2, {2}} PowerSet(A) = { emptySet, {1}, {2}, {{2}}, {1,2}, {1, {2}}, {2, {2}}, A } -------- --- --- ----- ----- -------- -------- - 1 2 3 4 5 6 7 8 The above shows the 8 elements of the PowerSet( {1, 2, {2}} ) *** Note that the cardinality of A is 3, 3 the cardinality of the PowerSet(A) is 2 So as you can see, sets can contain sets as members. The PowerSet of a set is every possible subset of that set. --------------------- -------------------------------------------------------------------------- Can a set like {a} be an element of a set? -------------------------------------------------------------------------- Suppose you have a health club you are running. You sell individual memberships and you sell family memberships. A family is A SET OF INDIVIDUALS, right!? --- The SET of memberships in your health club might be as follows, after a few hours of selling memberships: Memberships M = {Jim, Sally, Ann, Luke, {Tom, Mary Ann}, Allen, {Alex, Mabel, Lucy}, Ross, Emily, Rachel, {Ben, Adam, Hoss, Little Joe, Hop Sing}, Jewel, Prince, Carlos Santana, Roberta Flack, ... } Note that the Cartwright family is NOT an individual. It is a set of individuals. Cartwrights C = {Ben, Adam, Hoss, Little Joe, Hop Sing} C elementOf M is TRUE Hoss elementOf M is FALSE Hoss elementOf C is TRUE You let people in the door as paid members if they are on the list of individuals, or if they are in a family, and on the list of individuals (in the set listing for) in that family. You can be an elementOf Memberships M, or your family could be an elementOf Memberships M. Little Joe elementOf M is FALSE. Little Joe elementOf Cartwrights is TRUE Cartwrights elementOf M is TRUE > The PowerSet(A) would have 8 elements, its the set of ALL possible > > subsets of set A. For A= {1, 2, {2}} PowerSet(A) = { emptySet, {1}, {2}, {{2}}, {1,2}, {1, {2}}, {2, {2}}, A } -------- --- --- ----- ----- -------- -------- - 1 2 3 4 5 6 7 8 > > The above shows the 8 elements of the PowerSet( {1, 2, {2}} ) > > So as you can see, sets can contain sets as members. > > The PowerSet of a set is every possible subset of that set. > So how many odd elements--or are they subsets?--are there? I count 4 > ( {1}, {2}, {{2}}, and A (which has 3 elements). And how many even > elements? I count 3, ({1,2}, {1, {2}}, and {2, {2}). That's a total of > 7. > So what do we say about the null set? It has 0 elements. Is 0 even or > odd? Or do we just have 4 even, 3 odd, and 1 null? 0 is an even number!!!!!!!!!!!!!!!!!!!!!! So the null set has an even number of elements! 0 is even. 2 divides 0 since 0 can be expressed as 2 times some integer p, 0 = 2p for p = 0. Odds = { ..., -5, -3, -1, 1, 3, 5, ... } Evens = { ..., -4, -2, 0, 2, 4, 6, ... } I will try to clarify more during class today. Thanks for the questions and feedback. Mark