My discrete math professor gave an example stating that the following relation is transitive, reflexive, symmetric, and antisymmetric.
A = {a,b,c,d}
R = {(a,a), (b,b), (c,c), (d,d)}
I do not understand how it is transitive because by my apparently wrong understanding of transitive, there should be an element (a,b) in R.
I have a homework question that asks me to give an example of a relation R that is reflexive, symmetric, but not transitive using the set A = {a,b,c,d}.
With my apparently incorrect understanding of transitive, I thought that R = {(a,a)} would be reflexive, symmetric but not transitive.
I also have to give an example of a relation R that is reflexive, transitive, but not symmetric and not antisymmetric. I am completely lost on this one.
Thanks
There needn't be an $(a, b) \in \mathscr R$. But "if there is $(a, b)$ and $(b, c)$ in $ \mathscr R $ then $(a, c)$ must also be in $ \mathscr R $. Since there are no such worries in the relation given in the set the "if - then" implication is trivially satisfied because it does not need to be satisfied. Hope I helped.
The condition for a lake to be "balanced" is if there are toads in the lake then there must be an equal number of frogs. If there are no toads in the lake then what? It is still "balanced" since the condition is not invoked.
For an example just use what the condition gives you. You must have $(a, b), (b, c) \in \mathscr R$ but not $(a, c)$ for some triplet $(a, b ,c)$. The other one too is not difficult at all. I suggest you think about it.