I'm trying to write formally in logic that 2 distinct sets A and B are bijective to each other and so have an equal cardinal number. This is my attempt, i think my definiton of an injective function seems a bit odd as its just the reverse of my surjective one. Many thanks in advance if you can help i really appreciate it.
$$(\forall y \in B, ~ \exists x \in A, ~f(x)=y)\land(\forall x\in A, \exists y \in B, f(x)=y)$$
My method was first half defines surjective and the second half an injective function combining them gives bijection. My books on predicate logic dont use comma's is this allowed?
I think your second one of the two statements connected by "and" should rather be: $$\nexists x_1, x_2 \in A: (x_1 \neq x_2 \land f (x_1)=f (x_2))$$