My assignment asks us to prove or provide a counter example for
If R is symmetric, then Rc is symmetric.
I know that if R is symmetric, then (x,y) and (y,x) are both in R, but what I do not understand is what is R^c, what does R^c do to the relation? I believe that complement basically means "not", so does that mean the complement of R is NOT symmetric?
I can't seem to wrap my head around this concept, any help would be appreciated.
Thanks
A relation is usually implemented as a set of ordered pairs. Equality on $\mathbb{N}^{\geq 1}$, for instance, is implemented as $E = \{ (1,1), (2,2), (3,3), \dots \}$.
So the complement would be $$\{ \text{ordered pairs} \} \setminus E$$ That is, $$\{(1,2), (1,3), \dots, (2,1),(2,3), \dots\}$$
Suppose $R^C$ were not symmetric. Then there would be $(a,b)$ in $R^C$ such that $(b,a)$ is not in $R^C$. But for $(b,a)$ not to be in $R^C$ means just that $(b,a)$ is a pair which lies outside the complement of $R$ viewed as a set of ordered pairs; that is, it lies in $R$ viewed as a set of ordered pairs, and hence in $R$. Therefore by $R$'s symmetricity, $(a,b)$ must also lie in $R$, and this is a contradiction.