3. Consider a family P = {Pi, , Pn} of all partitions of a finite set A. Let R b

Question

3. Consider a family P = {Pi, , Pn} of all partitions of a finite set A. Let R be the following binary relation on P: where maxcard(Pi) is the maximum number of elements in partition classes of Pi e.g if A=(a,b,c,d) and P, = {{a),(b), {c, d)), maxcard(Pi) = 2; for P,-{{a,d),(b,c)), maxcard(PJ) =maxcard(Pi), so P, is in relation R with Pi. (a) Let card(A) = K. What is the number of equivalence classes of R? Explain why. (b) Let A = {a,b,c,d,e). What is the number of partitions in the equivalence class of R which contains the partition sHa, b, c), [d, e))? Explain why.

Explanation / Answer

(a) an equivalence class is a set of objects that are equivalent to each other . let us denote the set A = {1,2,3,...., K} . now for partition taking single element in a set we will get a equivalence class. again, taking partition with maximum 2 element in a set we will get a equivalence class. again, taking partion with maximum 3 element in a set we will get a equivalence class. so continuing this, how many equivalence class you will get ? exactly the number of element the set A have. hence there are K equivalence relation in R.  

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