Consider a relation R: A → A which is both symmetric and transitive.
The following proof shows that the relation is also reflexive: Take a ∈ A. If a ∼ b then b ∼ a by symmetry, and hence a ∼ a by transitivity. Therefore, the relation is reflexive."
Is the proof correct or not?
I think it is incorrect. I think it should be "For every a ∈ A, there exists b ∈ A that a ~ b and b ~ a.", not just "Take a ∈ A". After that we can use symmetry and transitivity to prove it is reflexive. That is what I think.
Am I right? Or what is the correct answer and the reason? THANKS!