Prove that if a relation R on a set A is reflexive, symmetric and antisymmetric, then $R=I_A$

Question

Prove that if a relation $R$ on a set $A$ is reflexive, symmetric and antisymmetric, then $R=I_A$

I know a relation is a set of ordered pairs and that $I_A = (x,x)$ but I have no idea how to do this problem

So $(x,x)R(x,x)$- reflexive

$(x,y)R(y,x)$-symmetric

$(x,y)R(y,x)$ implies $x=y$ -antisymmetric

Answer 1

You need to show two separate things:

1. $I_A\subseteq R$, i.e. you need to show that for every $x\in A$ you have $(x,x)\in R$.
2. $R\subseteq I_A$, i.e. you need to show that if $(x,y)\in R$ then $x=y$.

Let $x\in A$, then because $R$ is reflexive we have $(x,x)\in R$, so $I_A\subseteq R$.

Now let $x,y\in A$ and $(x,y)\in R$. Then because $R$ is symmetric you also have $(y,x)\in R$, but $R$ is antisymmetric so if $(x,y)\in R$ and $(y,x)\in R$ then $x=y$.

Hence $R=I_A$.