Date: Mon, 22 Oct 2001 19:33:15 -0500 (CDT) From: Mark Jacobson To: 810-080-01@uni.edu, 810-080-02@uni.edu Subject: HW #5 links... HW #5 is due on Wednesday. Please take advantage of the four new links found in the week 9 row, which is 10/22 to 10/26 row of the http://www.cns.uni.edu/~jacobson/c080.html class web page. These three pages are based on two different questions that have been asked late last night and late this afternoon regarding the HW #5 exercises. By the way: Wright 112 lab has a stapler. The mathematics department office has a stapler. The computer science department office has a stapler. {b} is an elementOf { a, b, {b}, { } } 1 2 3 4 Do you see it there? Its the 3rd of the 4 listed elements. Is {a} an elementOf { a, b, {b}, { } } ? I don't really see {a} in the list of 4 elements, do you? Is the emptySet an elementOf { a, b, {b}, { } } ? 1 2 3 4 That 4th element sure looks like the empty set to me, so the answer is YES, the { } is one of the 4 elements of { a, b, {b}, { } } Is { a } an elementOf { a, b, {b}, { } } ? NO! Is { a } a subset of { a, b, {b}, { } } ? YES! Is { b } a subset of { a, b, {b}, { } } ? YES! Is { {b} } a subset of { a, b, {b}, { } } YES! Is { {b} } an element of the PowerSet of { a, b, {b}, { } } ? YES! 3 How many sets are in the PowerSet of { a, b, {b}, { } } ? 2 = 8 subsets Can you list them? PowerSet( { a, b, {b}, { } } ) = { {}, { a }, { b }, { {b} }, { {} }, {a,b}, {a,{b}}, {a,{}}, {b,{b}}, {b,{}}, {{b},{}}, {a,b,{b}}, {a,b,{}}, {a,{b},{}}, {b.{b},{}}, {a,b,{b},{}} } A = { a, b, {b}, { } } SUBSET SUBSET PowerSet(A) = { CARDINALITY COUNT {}, 0 1 { a }, { b }, { {b} }, { {} }, 1 4 {a,b}, {a,{b}}, {a,{}}, 2 6 {b,{b}}, {b,{}}, {{b},{}}, {a,b,{b}}, {a,b,{}}, 3 4 {a,{b},{}}, {b.{b},{}}, {a,b,{b},{}} 4 1 } How symmetric those subset counts appear to be. 4 Notice that the sum is 16 which is 2 |A| Cardinality(A) Cardinality( PowerSet(A) ) is |PowerSet(A)| = 2 = 2 |A| = 4 |PowerSet(A) = 16, which is 2 to the 4th power... Enough review for now.... Mark