Proving R = I_X: Equivalence Relation and Function Homework Solution

In summary, the conversation discusses proving that a set with an equivalence relation and a function is equal to the identity function. The solution involves showing that the relation is reflexive, meaning each element is only related to itself, and therefore equivalent to the identity function.
  • #1
Danielm
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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.

Homework Equations

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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  • #2
I wonder why there's only reflexive needed. But seems ok.
 

1. What does R = I_X mean in this context?

In this context, R = I_X means that the relation R is equivalent to the identity relation I_X on a set X. This means that every element in X is related to itself, and no other elements are related to each other.

2. What is an equivalence relation?

An equivalence relation is a type of binary relation on a set that satisfies three properties: reflexivity, symmetry, and transitivity. This means that every element is related to itself, the relation is symmetric (if x is related to y, then y is related to x), and the relation is transitive (if x is related to y and y is related to z, then x is related to z).

3. How is R = I_X related to functions?

In this context, R = I_X means that the relation R can be represented as a function on the set X. This is because the identity relation I_X is a special type of function, where each element in the domain (X) is mapped to itself in the range.

4. How do you prove that R = I_X is an equivalence relation?

To prove that R = I_X is an equivalence relation, you must show that it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity. This can be done by showing that every element is related to itself, the relation is symmetric, and the relation is transitive for any elements in the set X.

5. Can you provide an example of R = I_X?

Yes, an example of R = I_X would be the relation "is equal to" on the set of integers. In this case, every integer is related to itself by the "is equal to" relation, but no other integers are related to each other. Therefore, R = I_X is satisfied.

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