Is this relation symmetric? I think it is but my professor claims it isn't.

Is the relation $R = \{(a,b),(a,c),(b,a),(b,c),(c,a),(c,b),(d,d)\}$ symmetric?

My professor claims that if $(d,d)$ was not included, it would be symmetric, but the inclusion of $(d,d)$ ruins it because $d$ has to connect to another element of the relation.

Solution 1:

A very simple way to see if a relation $R$ is symmetric is to check its inverse relation $R^{-1}$, where $(x,y)\in R\Leftrightarrow (y,x)\in R^{-1}$

Since for the given relation, $R=R^{-1}$, it is symmetric.

$\color{green}{\text{YOU ARE CORRECT.}}$

Solution 2:

Yes, that relation is symmetric. The definition of symmetry does not require each element to be connected to some other element; $R$ is symmetric iff for every $x,y$ such that $(x,y)\in R$ it is also the case that $(y,x)\in R$.

Solution 3:

One way to see if a relation is symmetric is to draw the table what is in relation to what:

$$\begin{array}{r|c|c|c|c}R&a&b&c&d\\\hline a&F&T&T&F\\b&T&F&T&F\\c&T&T&F&F\\d&F&F&F&T\end{array}$$

and just see if the table is symmetrical across the main diagonal.

And - in this case it is. Thus, the relation is symmetric. Having $(d,d)$ in it has nothing to do with it, it would be symmetric either way (i.e. whether the entry for $(d,d)$ is "true" or "false").