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Equivalence Classes

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Let P, P' be equivalence relations on a set A. Let n, n' be the number of equivalence classes of p, p', respectively.
A) define an equivalence relation p'' as follows:
xp''y <=> (xpy) and (xp'y)
what is the least number of equivalence classes of p''? What is the greatest number of equivalence classes of p''?

B)define an equivalence relation p''' as follows:
xp'''y <=> (xpy) or (xp'y)
what is the least number of equivalence classes of p'''? What is the greatest number of equivalence classes of p'''?

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Solution Summary

Equivalence classes are found from equivalence relations. The greatest number of equivalence classes of p is determined.

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A.)
Because, in set theory, and means intersection. Therefore, the number of equivalence classes of p'' will be equal to the intersection of n and n' classes.
Hence, if all n (of p) and n' (p') equivalence classes are completely disjoint, in that case the number of equvalence classes of p'' will be ...

Solution provided by:
Amrit Lal Ahuja, PhD (IP)
Education
  • BEng, Allahabad University, India
  • MSc , Pune University, India
  • PhD (IP), Pune University, India
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