Proving R = I_X: Equivalence Relation and Function Homework Solution

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SUMMARY

The discussion centers on proving that an equivalence relation R on a set X, which also functions as a function, is equivalent to the identity function I_X. The proof establishes that R must be reflexive, meaning for every element b in X, the relation holds only for b itself, confirming that R assigns each element x in X to itself. Thus, it is concluded that R = I_X, as R cannot relate any element to another distinct element.

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Homework Statement


Let X be a set and R ⊂ X × X. Assume R is an equivalence relation and a function. Prove that R = I_X, the identity function.

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The Attempt at a Solution


Proof

We know that R has to be reflexive, so for all elements b in X, bRb but b can't be related to any other element because of the definition of function, so b is just related to b. It's easy to see that the relation is equivalent. Therefore, R=I_x because R assigns to each element x in X, the element x in X.

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I wonder why there's only reflexive needed. But seems ok.
 

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