Injectiveness, surjectiveness, biuniqueness. Countability and uncountability of sets

Mapping is injective, if every y ∈ Y_f has unique inverse image, that is from equality

f(x₁) = f(x₂) → x₁ = x₂.

Mapping is surjective, if Y = Y_f , that is every y ∈ Y has inverse image.

Mapping is biunique, if it is injective and surjective.

Biunique mapping realizes one-to-one correspondence between sets X and Y.

Sets, connected by biuniqueness, are called equipotent.

Sets, equipotent to the set of natural numbers, are called countable.

One-to-one mapping has inverse.