# Special Binary Relations/ Empty Relation, Universal Relation And identity Relation?

The universal relation U = A × A.
(Correct me if I'm Wrong). I believe that the Universal Relation is an Equivalence Relation
The empty relation E = ∅.
From my understanding, a Empty relation on a non empty Set A Symmetric, Anti Symmetric and Transitive.
The identity relation Id = {(x, x) : x ∈ A}
I'm Having Trouble with this one. I'm not too sure if this relation is Transitive, Symmetric or antisymmetric? I'm certain this is reflexive tho.
A is a set of integers

• The empty relation is not reflexive (i.e., it's not true that $(x,x)\in E$ for every $x$). The identity relation is actually the smallest equivalence relation on any given set. Hint for all of this: check the definitions for reflexive, symmetric, antisymmetric, and transitive. If you know exactly what these mean, these questions should be easy. – Kyle Miller Dec 9 '14 at 4:15
• oops, That was a typo. Yes i'm aware that the empty set is not reflexive. – Cheng Dec 9 '14 at 4:18
• What do you mean by the smallest equivalence relation? – Cheng Dec 9 '14 at 4:21

Here's a way to show the identity relation is transitive. Suppose $(a,b)\in\mathrm{Id}$ and $(b,c)\in\mathrm{Id}$. Since then $b=a$ and $c=b$, then $a=c$, and so $(a,c)\in\mathrm{Id}$, too. (In other words, if $a\sim b$ and $b\sim c$ then $a\sim c$, where $\sim$ stands for the identity relation.)
Symmetry is saying that when $(a,b)\in\mathrm{Id}$ then, since $b=a$, then $(b,a)\in\mathrm{Id}$, too.
When I said that it is smallest, what I mean is that 1) it is an equivalence relation and 2) for any other equivalence relation $R\subset A\times A$ that $\mathrm{Id}\subset R$ (that is, the identity relation is always lurking inside any other equivalence relation).