TITLE: Is this relation symmetric? I think it is but my professor claims it isn't.
QUESTION [10 upvotes]: 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.

REPLY [2 votes]: 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").