# Thread: getting started on a relation question

1. ## getting started on a relation question

I just need help getting started on this question. the first line (or two) is eluding me:

Let R be a relation on A and suppose that R is symmetric and transitive. Prove thefollowing:

If, for all xE A, there is yE A such that xRy, then R is an equivalence relation.

where xE A and yE A mean x and y are elements of A.

I know i have to prove reflexitivity, and I only have this written down...my mind has stopped working:

Let xE A, then R is an equivalence relation
=> (nothing written here...i'm stumped)

any help would be appreciated. Thank you

2. Hi
What you have to do is assume that $\displaystyle \forall x\in A,\ \exists y\in A\ \text{s.t.}\ x\mathcal{R}y$ and, with that hypothesis and the fact that $\displaystyle \mathcal{R}$ is symmetric and transitive, prove that $\displaystyle \mathcal{R}$ is an equivalence relation.

As you said, you just have to prove that $\displaystyle \mathcal{R}$ is reflexive.

So let $\displaystyle x$ be an element of $\displaystyle A,$ the hypothesis says that there is a $\displaystyle y\in A$ such that $\displaystyle x\mathcal{R}y.$
But, since $\displaystyle \mathcal{R}$ is symmetric, what relation can we find using only that $\displaystyle x\mathcal{R}y.$ Finally, use the transitivity to prove that $\displaystyle x\mathcal{R}x.$