Let A 1, A 2 , A 3...., A s be a partition of a set X. Define a relation R on X

Question

Let A1, A2, A3...., As be a partition of a set X. Define a relation R on X by xRy if and only if x and y belong to the same part of the partition. Prove that R is an equivalence relation? (I would really appreciate the help in the first part. Second can be done after I figure out the first part) Let A1, A2, A3...., As be a partition of a set X. Define a relation R on X by xRy if and only if x and y belong to the same part of the partition. Prove that R is an equivalence relation? (I would really appreciate the help in the first part. Second can be done after I figure out the first part)

Explanation / Answer

I am not sure what you mean by the first part. The question is this: There is a set X which has been partitioned into s parts A1,A2,...As. This partition is fixed. Now define xRy for x, y in X if and only if x,y are in the same part A_i for some i. This is the definition of the relation. I hope this point is clearly understood. What you now to do is prove that this relation is an equivalence relation. This means, showing that this relation R is reflexive, symmetric, and transitive. It is clearly reflexive; xRx since x and x DO lie in the same part (namely the part containing x). Also, if xRy then yRx since if xRy then x,y are in the same part. Clearly then y, x are in the same part. For transitivity, suppose xRy, yRz. Then x,y lie in some A_i. Since yRz, y,z are in the same part A_j for some j. But since the A_i's partition the set X, two distinct sets A_i,A_j have empty intersection unless i = j. Hence we must have that i = j, i.e., x,y z all lie in A_i. This clearly implies that xRz which proves transitivity.

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