Does every mathematical operation has an inverse operation?

Question

For example, we say that the addition and subtraction are inverse operations like that does each and every mathematical operation has an inverse operation?

Answer 1

No. If you take the next "immediate" operation, multiplication, note that it doesn't always have an inverse, since division is not always defined. For instance: $$0\cdot 40=0\Rightarrow \frac{40}{0}=?$$ Generically speaking (and this is by no means a theorem or an exact statement, only an intuition), "most" operations are not invertible, since you only need one element to not have an inverse in order to ruin everything.

Answer 2

Addition and subtraction have inverses if you only allow a single operand to vary (and fix the other one). But if you think of these operations as mapping an ordered pair of numbers onto a single number, then these operations don't have an inverse.