DM - Functions & Relations -Q10

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An equivalence relation on a set A partitions the set A into non-empty disjoint subsets such that each element of A belongs to exactly one of these subsets. The set of these  non-empty disjoint subsets of A, that collectively contain all elements of set A, is called a Partition of set A.

A partition P1 of a set A is called a refinement of the partition P2 of set A if every set in P1 is a subset of exactly one of the sets in P2.

Consider the following statements :

S1: Let B be the set of all bit strings of length 16. Then the partition of B formed by equivalence classes of bit strings that agree on the last eight bits is a refinement of the partition of B formed from the equivalence classes of bit strings that agree on the last four bits.

S2: The partition of set of all positive integers formed from congruence classes modulo 6  is a refinement of the partition formed from congruence classes modulo 3.

Which of the following statements is true?

(A). Only S1

(B). Only S2

(C). Both S1 and S2

(D). None of the above
asked Jun 24 in Discrete Maths by gbeditor (44,560 points)
reshown Jun 27 by gbeditor

1 Answer

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Best answer

First, try to understand what is refinement. 

A partition P1 is called a refinement of the partition P2 if every set in P1 is a subset of one of the sets in P2.

S1: The partition of the set of bit strings of length 16 formed by equivalence classes of bit strings that agree on the last eight bits is a refinement of the partition formed from the equivalence classes of bit strings that agree on the last four bits

Here in P1  for each distinct pattern of last 8 bits one partition is created.  In P2 for each distinct pattern of last 4 bits one partition is created. 
Every partition of P1 is a subset of some partition of P2, so P1 is refinement of P2. 
In same way in S2 too P1 is refinement of P2. 

answered 5 days ago by deepak-gatebook (112,760 points)
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