1. Let A be the set of all (nonempty) subsets of integer numbers between 1 and...

 

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1. Let A be the set of all (nonempty) subsets of integer numbers between 1 and K (including 1 and K), K > 1. Let R be a binary relation on A defined as follows: ARA, sum (A) = sum(Aj) where A; and A; are elements of A, and sum(A) denotes the sum of all elements of Ai, so, for example, {3,7} and {1,2,3,4} are in relation R because 3+7=1+2+3+4. Observe that R is an equivalence relation on A. (a) What is the number of equivalence classes of R (as a function of K)? Explain why. (b) Assume K > 10. List all subsets in the equivalence class which contains {10}. (c) How many equivalence classes contain just one element? What are these classes? Explain why. 1. Let A be the set of all (nonempty) subsets of integer numbers between 1 and K (including 1 and K), K > 1. Let R be a binary relation on A defined as follows: ARA, sum (A) = sum(Aj) where A; and A; are elements of A, and sum(A) denotes the sum of all elements of Ai, so, for example, {3,7} and {1,2,3,4} are in relation R because 3+7=1+2+3+4. Observe that R is an equivalence relation on A. (a) What is the number of equivalence classes of R (as a function of K)? Explain why. (b) Assume K > 10. List all subsets in the equivalence class which contains {10}. (c) How many equivalence classes contain just one element? What are these classes? Explain why.