- #1

evinda

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We are given the set $E=\{d,e,f \}$, $d,e,f$ different from each other and the relation $I_{E}=\{ <x,x>: x \in E\}$. Prove that $I_{E}$ is a set. In addition, show that the relation $I_{E}$ is an equivalence relation in $E$ and find all the equivalence classes.

That's what I have tried:

We define $\phi(x)=<x,x>: x \in E$.

Then, we have that $\forall <x,x>(\phi(x)) \rightarrow <x,x> \in E \times E$.

$E \times E$ is a set.

So, from the theorem: "Let $\phi$ a type. If there is a set $Y$, such that $\forall x(\phi(x)) \rightarrow x \in Y$, then there is the set $\{ x: \phi(x) \}$", we have that $I_{E}=\{ <x,x>: x \in E\}$ is a set.

The relation $I_{E}$ is an equivalence relation:

- reflective: $<x,x> \in I_{E} \rightarrow <x,x> \in I_{E}$
- symmetric: $<x,y> \in I_{E} \rightarrow x=y \rightarrow <y,x> \in I_{E}$
- transitive: $<x,y> \in I_{E} \wedge <y,z> \in I_{E} \rightarrow x=y \wedge y=z \rightarrow <x,z> \in I_{E}$