# Thread: Symmetric relation. Cannot understand.

1. ## Symmetric relation. Cannot understand.

I would guess the relation R would be symmetric if A was equal to {b,c,d}, but not {b,c,d,e} as in the textbook?

But let me try to reason through it.

Let's say we consider eRb. Then if the relation is symmetric, it must follow that bRe. But since there's no (e,b) in R (i.e. ~eRb), the statement $\forall x,y \in A, xRy \implies yRx$ is vacuously true for x=e, y=b, and so with any other pair involving e. Hence R is symmetric. Is that correct?

2. ## Re: Symmetric relation. Cannot understand.

Originally Posted by maxpancho

I would guess the relation R would be symmetric if A was equal to {b,c,d}, but not {b,c,d,e} as in the textbook?
But let me try to reason through it.
Let's say we consider eRb. Then if the relation is symmetric, it must follow that bRe. But since there's no (e,b) in R (i.e. ~eRb), the statement $\forall x,y \in A, xRy \implies yRx$ is vacuously true for x=e, y=b, and so with any other pair involving e. Hence R is symmetric. Is that correct?
Here are some quick tests.
Refelcxive means that $\displaystyle \Delta¬_{\mathscr{R}}\subseteq\mathscr{R}$
Symmetric means that $\displaystyle \mathscr{R}=\mathscr{R}^{-1}$
Transitive means that $\displaystyle \mathscr{R}\circ\mathscr{R}\subseteq\mathscr{R}$

3. ## Re: Symmetric relation. Cannot understand.

Thanks. Although I don't understand the notation quite that well yet (in regard to analysis) and more concerned with the particular paragraph for now.

But okay if no one's going to correct me, then I will assume I'm right.