Need help on partitiions homework. Let P be a Partition of a set A. Suppose that

Question

Need help on partitiions homework.

Let P be a Partition of a set A. Suppose that |P| = 6 and that three elements of P each have 5 elements and that the other three each have 3 elements. How many elements does A have? Now suppose that P is a partition of a set A such that |P| = 15 and such that for each S P . |S| = 7. What is |A|? On Z define the relation where m n means that m and n are both even or that they are both odd. Quickly check that this is an equivalence relation on Z. Now find the associated partition P of Z Finally explain why the partition P of Z is compatible with the operation + on Z. Extra Credit Let A be a set and suppose that is an equivalence relation on A. Let P be the associated partition of A. Now let * he an operation on A. We know what it means to say that * is compatible with P. Find out what this means in terms of and *. So the problem is to find out what has to be true about and * together (without referring to P) so that P and * are compatible.

Explanation / Answer

1)Since Partition of a set is collection of disjoint subsets.Therefore no of elements of A are 3*5+3*3=24...ANS 2) following similar argument |A|=15*7=105....ANS 3)for the relation to be equivalent it has reflective,symmetric and transtive..Now(let the relation be R) mRm always holds true since a number is either even or odd..so reflexive..Again, if mRn then m,n are both odd or even,hence nRm is also true..thus symmetric..Finally if mRn and nRp then; m,n and n,p are both odd or even ==>m,p are both odd or even==> mRp holds true..hence transitive.,Therefore equivalent..ANS

3b) P can be set of disjoint sets ofthe union of which is Z. eg; {1},{2},... Qualitatively since this include only disjoint sets ,hence P is compatiblw with + operation on Z (EXTRA CREDIT) In simple terms if m=~n is true and m*n=p then m=~p must be true so that P and * are compatible

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