I have one predicate and two finite sets A and B.
$$\mathbf{ \text{I(x): x is an injective function.}}$$
From these I construct the quantified statement:
$$\mathbf{ \forall x\in\text{U:[ I(x)]}}$$
$$\mathbf{ \text{U = \{ f is a function | f: A}\mapsto}\text{B\}}$$
What assumptions do I need to make on each of the sets to make each of the quantified statements true?
The answers in my book says the first statement is true when |A| = 0, |A| = 1, or |B| = 0, though I'm not sure I follow the logic.
If |A| = 0 then I can see that the statement is true in regards to the only function being the empty function which I've read is injective (though haven´t understood exactly why yet).
But if |A| = 1, how can I assume f(x) will have an inverse without making any assumptions about B?
Also if |B| = 0, is this statement just vacuously true with regards to injectivity?
Best Answer
A function $f$ is injective if $x \neq y \Rightarrow f(x) \neq f(y)$. So if $\operatorname{dom}(f)=A$ and $\vert A \vert \leq 1$, then $f$ must always be injective because you can never have $x, y \in A$ with $x \neq y$ so the implication is vacuously true.