Proving R = I_X: Equivalence Relation and Function Homework Solution

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R is defined as an equivalence relation and a function on set X. The proof establishes that R must be reflexive, meaning each element b in X relates only to itself, as it cannot relate to any other element due to the function's definition. This leads to the conclusion that R assigns each element x in X to itself, confirming that R equals the identity function I_X. The discussion raises questions about the necessity of only proving reflexivity, but the reasoning appears sound. The conclusion is that R = I_X holds true under the given conditions.
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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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