In my uni script, a bijective function is defined as a function that is simultaneously injective and surjective. However, while working through some set-theoretic proofs, I have also encountered a (to my understanding) different definition - that a function is bijective if domain and range have the same cardinality. Intuitively, it makes sense, but formally - why?
EDIT: For clarity, the definitions of injectivity and surjectivity I know from uni are of the form $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$ and $\forall y\in Y \exists x\in X:y = f(x)$.