State the smallest relation containing the relation $$\{(1,2),(2,1),(2,3),(3,4),(4,1)\}$$ that is:
a) reflexive and transitive.
b) reflexive, symmetric and transitive.
For me reflexive would be something like: $\{(1,2),(1,2)\}$; transitive would be like $\{(1,1)\}$ or $\{(3,1)\}$ but these two are not in the relation. Therefore, I don't know how to get the one that is both reflexive and transitive. As well as one that has all three properties. I have to admit that I am kind of lost here.
If we let $X = \{(1,2),(2,1),(2,3),(3,4),(4,1)\}$, then the smallest relation on $X$ that is reflexive, symmetric, and transitive is $\{(1,2),(2,1)\}$. Note that for a relation to be smaller and symmetry to be satisfied, you would have to have a relation of form $\{(x,x)\}$. The same goes for the smallest relation that is reflexive and transitive.