Let $A$ be any set, and let $I: A\to A$ be the identity function on $A$. To show this identity function over $A$ is a bijection. We can show that it is injective and surjective. We can readily know that it is surjective because the identity relation is reflexive. Yet, in order to show that it is injective, should we appeal to both the symmetry and transitivity of the identity function? I can see the point that it should be symmetric in order to be injective, but I don't see why it has to be transitive.
[Math] Show that the identity function over any set is a bijection
functions
Best Answer
The identity relation is reflexive and a function and that is enough to prove bijectivity the way you want to do it. Every x is mapped to itself (reflexivity) and to nothing else - since it is a function - only one mapping for equal inputs. The only reason for equal mappings here are equal inputs, hence it is injective too.