Question

Let f be a function from A into X, and let Y, Z be proper subsets of X. How do I prove
the inverse function of X = A and
the inverse function of (Y') = the inverse function of (Y)"

Answer 1

Let $f$ be a function from $A$ into $X$, and let $Y, Z$ be proper subsets of $X$. How do I prove the inverse function of $X = A$ and the inverse function of $(Y') =$ the inverse function of $(Y)''$.

As there is a function

$$g(A): A \to X$$

There must be inverse function

$$f(X) = g^{-1}(A): X \to A$$

As $Y$ is a subset of $X$ then

$$f(Z) = g^{-1}(A): Z \to A, Z \in X$$ $$f(Y) = g^{-1}(A): Y \to A, Y \in X$$

But as

$$Y, Z \in X$$

Then $f(Z)$ and $f(Y)$ exist.