Let $A={a,b,c,d,e,f}$ and let $R\subset A\times A$ be a relation which is symmetric and transitive. You have been given some partial information about the relation which is that the following are known to be true for the relation:
$$R(e,a), R(b,f), R(e,c), R(d,a), R(b,a)$$
Given all the above information does $R(c,f)$ hold or not? Explain your answer using the information you have been provided with above.
I would argue that $R(c,f)$ does not hold. No two pairs would equal $R(c,f)$ and $R(f,c)$.
Does this seem correct?
Update: Just thought of this.
R(b,f) and R(b,a) would be the same as R(f,a).
R(f,a) and R(e,a) would be the same as R(f,e).
Therefore, R(f,e) and R(e,c) would be the same as R(f,c).
As the relation is symmetric, R(f,c) -> R(c,f).
How's that?