Simple problem of relations which are symmetric

Question

I have a question regarding relations that are symmetric, drawn from an algebra book I am busy with. It states...

Let $R$ be a relation from $X\,to\, X$

Let $ \hat R$ be the converse of $R$.

Then, $R$ is symmetric, if and only if $ \hat R \subset R $.

My problem is for what reason is the equality sign not used instead of the subset sign as this seems more logical to me, as in rather...

$R$ is symmetric, if and only if $ \hat R = R $

Answer 1

Suppose that $\hat{R}\subset R$. Take $a,b\in X$ such that $a\mathrel{R}b$. Note that\begin{align}a\mathrel{R}b\iff&b\mathrel{\hat{R}}a\\\implies&b\mathrel{R}a\text{ (since $\hat R\subset R$)}\\\iff&a\mathrel{\hat R}b.\end{align}So, it follows from $\hat{R}\subset R$ that $R\subset\hat R$ and therefore that $R=\hat R$.

Answer 2

$R=\{(x,y)| x,y\in X\}$

$\hat R=\{(y,x)|(x,y)\in R\}$

I understand you mean $R$ is symmetric if and only if $\hat R\subset R$ to be that

$R$ is symmetric if and only if $\hat R=R$.

But $R$ is symmetric if and only if $\hat R=R$ is not meaningful. It is obvious.

• $R$ is symmetric if and only if $\hat R\subset R$ this fact say: Subset is sufficient condition. Maybe, we can have some elements in $R$ not in $\hat R$. Again first direction is obvious. But you need to prove second direction.

$\Longleftarrow$ Take $(x,y)\in R$ arbitrary. Then we must show that $(x,y)\in \hat R$.

$(x,y)\in R\Longrightarrow (y,x)\in \hat R\Longrightarrow (y,x)\in R\Longrightarrow (x,y)\in \hat R$