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From definition of an equivalence relation it easily can be proved that an equality relation denoted "$=$" is indeed an equivalence relation, it means that every two arbitrary elements on a set $X$ with respect to $=$ are related to each other if they have the same value, therefore the set created by "$=$" on $X$ contains ordered pairs $(x,x)$ where $x \in X$.

On the other hand an identity relation over $X$ (which is a homogeneous binary relation ) denoted $\text{Id}_{X}$ is also contains ordered pairs $(x,x)$ where $x \in X$.

I think equality is a special case of an identity relation, for example we say "equality is the finest equivalence relation" I think it's true because an equality equivalence relation is actually an identity relation, and since identity relation is contained in all of equivalence relations hence equality is also contained in all of them, but can we say that :

is Every arbitrary identity relation also an equivalence relation?

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    How do you define "identity relation"? I can't imagine anything other than saying it satisfies exactly the same properties that an equivalence relation does. – lulu Feb 02 '20 at 15:47
  • $Id_{X}=\left{\left(x,x\right):x \in\ X\right}$ –  Feb 02 '20 at 15:48
  • Well, isn't it obvious that that satisfies the requirements of an equivalence relation? – lulu Feb 02 '20 at 16:03
  • @lulu, is equality equivalence relation equal to identity relation ? (where they both are defined on a same set) and assuming $A=\left{1,2,3\right}$ then what is its identity relation? –  Feb 02 '20 at 16:06
  • Again, this all depends on what you mean. I'd argue that every equivalence relation was an identity relation, though of course that depends on what underlying set you mean. I think all these potential ambiguities go away if you write out exactly what you mean, with full definitions. – lulu Feb 02 '20 at 16:08
  • @lulu Given a set $X=\left{x_{1},...,x_{n}\right}$,then by definition every equivalence relations $R$ on $X$ must be reflexive , means $\forall x_i \in X$ the ordered pair $\left(x_i,x_i\right)$ is in $R$ , so $Id_{X}$ is in all of the equivalence relations defined on $X$. if we define the equivalence $R$ on $X$ to be equality relation $=$, then the set created by this relation would be in the form $\left{\left(x_{i},x_{i}\right):\ x_{i}\in\ X\right}$ which is exactly the definition of $Id_{X}$.I think an equality equivalence relation and $Id$ defined on a same set are the same. –  Feb 02 '20 at 16:30
  • Well, with that definition then they are not the same. The set $X$ admits many equivalence relations. For instance, $x_i\sim x_j,\forall,i,j$ is an equivalence relation. But that is not the identity relation on $X$. – lulu Feb 02 '20 at 16:43
  • @lulu, so what is the condition that an equivalence relation is equal to an equality equivalence relation? –  Feb 02 '20 at 17:50
  • Once again: please edit your post to include all the definitions of the terms you are using. I don't think there is any controversy here, just a semantic distinction. – lulu Feb 02 '20 at 19:02

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It is easy to see that the "identity relation" and "equality" relation are the same. To say that the identity relation contains only those elements of type $\left( x, x \right)$, is the same as saying that $x$ is related to $y$, through the identity relation if and only if $x = y$.

Since you have already proven that equality relation is an equivalence relation, the identity relation also becomes an equivalence relation.

Aniruddha Deshmukh
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  • consider a set $A=\left{1,2,3\right}$ , then $$Id_{A}=\left{\left(a,a\right):a \in\ A\right}=\left{\left(1,1\right)\left(2,2\right)\right}$$ but $$\sim_==\left{\left(1,1\right)\left(2,2\right),\left(3,3\right)\right}$$ –  Feb 02 '20 at 15:53
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    When you say $\left\lbrace \left( a, a \right) : a \in A \right\rbrace$, it means that you have to collect all those ordered pairs $\left( a, a \right)$, that can be formed by varying $a$ in the set $A$. Therefore, in this case, the set is equal to $\left\lbrace \left( 1, 1 \right), \left( 2, 2 \right), \left( 3, 3 \right) \right\rbrace$, exactly the same as equality you have defined. – Aniruddha Deshmukh Feb 02 '20 at 16:11
  • Every equivalence relation on a set $X=\left{x_{1},...,x_{n}\right}$ is reflexive by definition, identity relation is also reflexive and vacuously is symmetric/transitive, hence an equivalence relation, also an equality relation is reflexive and since any two elements are related to each other if they have a same value implies any ordered pairs should be in the from $\left(x_{i},x_{i}\right)$ , since in a set repetition does not add a new member, therefore elements of an equality relation are all in the form $\left(x_{i},x_{i}\right)$, which is exactly the same as the identity relation on $ –  Feb 03 '20 at 08:28
  • Am I right? also I think it would be better to define an identity relation $\text{Id}⊆X×X$ such that: $${\text{Id}_X}:=\left{\left(x,x\right):\color{red}{∀}x\ ∈\ X\right}$$ –  Feb 03 '20 at 08:32