Explanation Given information: The relation R is an equivalence relation on the set A. X = {-1, 0, 1} and A = P(X). R is defined on P(X) as follows: For all sets s and r in P(X), S R T ⇔ the sum of the elements in S equals the sum of the elements in T Calculation: X = { a , b , c } A = P ( X ) R = { ( S , T ) ∈ A × A | Sum of elements in S is the same as sum of elements in T } Let us first determine all elements in A, which are thus all subsets of X : A = { 0 , { − 1 } , { 0 } , { 1 } , { − 1 , 0 } , { − 1 , 1 } , { 0 , 1 } , { − 1 , 0 , 1 } } Two elements are then related if the sum of the elements of the sets is the same: R = { ( ϕ , ϕ ) , ( ϕ , { 0 } ) , ( ϕ , { − 1 , 1 } ) , ( ϕ , { − 1 , 0 , 1 } ) , ( { 0 } , ϕ ) , ( { 0 } , { 0 } ) , ( { 0 } , { − 1 , 1 } ) , ( { 0 } , { − 1 , 0 , 1 } ) , ( { − 1 , 1 } , ϕ ) , ( { − 1 , 1 } , { 0 } ) , ( { − 1 , 1 } , { − 1 , 1 } ) , ( { − 1 , 1 } , { − 1 , 0 , 1 } ) , ( { − 1 , 0 , 1 } , ϕ ) , ( { − 1 , 0 , 1 } , { 0 } ) , ( { − 1 , 0 , 1 } , { − 1 , 1 } ) , ( { − 1 , 0 , 1 } , { − 1 , 0 , 1 } ) , ( { − 1 } , { − 1 } ) , ( { − 1 } , { − 1 , 0 } ) , ( { − 1 , 0 } , { − 1 } ) , ( { − 1 , 0 } , { − 1 , 0 } ) , ( { 1 } , { 1 } ) , ( { 1 } , { 1 , 0 } ) , ( { 1 , 0 } , { 1 } ) , ( { 1 , 0 } , { 1 , 0 } ) } 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 8.3, Problem 9ES

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To find the distinct equivalence classes of R.

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Ch. 8.1 - If R is a relation from A to B, xA , and yB , the...Ch. 8.1 - Prob. 2TY Ch. 8.1 - Prob. 3TY Ch. 8.1 - Prob. 4TY Ch. 8.1 - If R is a relation on a set A, the directed graph...Ch. 8.1 - As in Example 8.1.2, the congruence modulo 2...Ch. 8.1 - Prove that for all integers m and n,m-n is even...Ch. 8.1 - The congruence modulo 3 relation, T, is defined...Ch. 8.1 - Define a relation P on Z as follows: For every...Ch. 8.1 - Prob. 5ES

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To find the distinct equivalence classes of R.

Solution Summary: The author explains that R is an equivalence relation on the set A. X = -1, 0, 1 and A = P(X).

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To find the distinct equivalence classes of R.

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Chapter 8 Solutions