# Set theory, functions and inverses

I'm doing an intro course on set theory and have the question if the inverses of the surjective functions in the sets $$A=\{a, b\}$$ and $$B= \{c, d, e\}$$ are also functions.

So far, I thought that the inverse of a function $$f(x)$$ for example, is $$f^{-1}(y)$$, meaning that every function also has an inverse (which is also a function). Given that this would make the question rather redundant, I'm not quite sure in my assumption anymore, so I would be glad if someone could verify or falsify (and explain it properly) it.

• Inverse defined from the range of the given function is a function. – Thomas Shelby Nov 19 '18 at 15:11

A relation is a set of ordered pairs, and it's inverse is the following relation: $$R^{-1}:=\{(y,x)|(x,y)\in R\}$$ And $$R$$ is called a function if $$\forall x,y_1,y_2$$, $$(x,y_1)\in R \land (x,y_2)\in R \implies y_1=y_2$$. In the case of functions, we can always define a formal inverse, and we call a function $$f$$ injective if it's formal inverse is also a function. And not all of the surjections are injections, for example let $$f:=\{(a,c),(b,d),(a,e)\}$$ Now, $$f$$ is a surjection from $$A$$ to $$B$$, but it's not an injection, because it's inverse is not a function.