I'm trying to understand how to write predicate logic describing a function that is not onto.
A function is onto if every element in the codomain gets mapped to by some element in the domain using the given function.
The function $\: f:\mathbb{R}\rightarrow\mathbb{R}, \: f(x)=\lceil x\rceil$ is not onto; All codomain elements are not mapped to.
To my best understanding the predicate logic for this function in relation to being onto would look like:
$$ \forall x\neg\exists y\{(x,y)\in\mathbb{R}\mid f(x)=y\not=noninteger\} $$
- Is this the correct form?
- How would I write "y can not be a fraction" mathematically?
If you want to say the function is not onto (ignoring what $f$ is), you can say $\exists x \lnot \exists y f(y)=x$, which just says there is some number $x$ that is not the image of any $y$. To say $y$ is an integer you can say $y \in \Bbb Z$, but if you are being formal and operating in $\Bbb R$ you can't identify the integers within $\Bbb R$.