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gbean
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Homework Statement
Suppose A[tex]_{\lambda}[/tex], [tex]\lambda[/tex] in L, represents a partition of the nonempty set A. Define R on A by xRy <=> there is a subset A[tex]{\lambda}[/tex] such that x is in A[tex]{\lambda}[/tex] and y is in A[tex]{\lambda}[/tex]. Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets A[tex]{\lambda}[/tex].
Homework Equations
Equivalence relation: ~ that is reflexive, symmetric, and transitive.
Partition: Mutually disjoint sets that divide up the parent set; the union of the partition is the whole set
Equivalence classes: AKA the partition sets
The Attempt at a Solution
To be an equivalence relation, it has to be reflexive, symmetric, and transitive.
xRx: There is a subset A[tex]_{\lambda}[/tex] such that x and x are in it.
Assume xRy. There is a subset A[tex]_{\lambda}[/tex] such that x and y are in it. It follows that yRx.
Assume xRy, yRz. There is a subset A[tex]_{\lambda}[/tex] such that x and y are in it. There is a subset such that y and z are in it. It follows that xRz.
I'm not sure how to prove that the equivalence classes are the subsets A[tex]_{\lambda}[/tex]? Isn't that just by definition, that the partition sets are the equivalence classes?