Explanation Given information: The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. X = { a, b, c } and A = P(X). R is defined on A as follows: For all sets U and V in P(X), U R X = {a, b, c} and A = P ( X ) . R is defined on A as follows: For all sets U and V in P ( X ), U R V ⇔ N ( U ) = N ( V ) . (That is, the number of elements in U equals the number of elements in V .) Calculation: X = { a , b , c } A = P ( X ) R = { ( B , C ) ∈ A × A | B has the same number of elements as C } Let us first determine all elements in A, which are thus all subsets of X : A = { ϕ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c } } Two elements are related if they contain the same number of elements: R = { ( ϕ , ϕ ) , ( { a } , { a } ) , ( { a } , { b } ) , ( { a } , { c } ) , ( { b } , { a } ) , ( { b } , { b } ) , ( { b } , { c } ) , ( { c } , { a } ) , ( { c } , { b } ) , ( { c } , { c } ) , ( { a , b } , { a , b } ) , ( { a , b } , { a , c } ) , ( { a , b } , { b , c } ) , ( { a , c } , { a , b } ) , ( { a , c } , { a , c } ) , ( { a , c } , { b , c } ) , ( { b , c } , { a , b } ) , ( { b , c } , { a , c } ) , ( { b , c } , { b , c } ) , ( { a , b , c } , { a , b , c } ) } Let us next group the ordered pairs in R that have at least one common element (with at least one of the other elements in the group)

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Chapter 8 Solutions

Discrete Mathematics With Applications

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To find the distinct equivalence classes of R.

Solution Summary: The author explains that the relation R is an equivalence relation on the set A.

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