A question about notation regarding functions…

I've been trying to wrap my head around this for a while now, but am having trouble understanding something. This is what the textbook says: In the first highlighted region, what does the k (arrow) f(k) mean? I've never come across this notation before, and I can't even google it because I don't know what it's called.

Also, what does it mean when it says "the class of all cycles" in the second region.

• k\mapsto f(k) $k\mapsto f(k)$ means that the element $k$ in the domain maps to the element $f(k)$ in the codomain under the action of the function $f$. – JMoravitz Dec 2 '16 at 1:03

The $\mapsto$ arrow is called "maps to" and it is a way to define a function (or other mapping) without giving it a name. $x \mapsto f(x)$ is like saying there exists a function $g$ such that $g(x) = f(x)$ without the additional bookkeeping of adding a new function name. Alternatively it can be read as a lamba function $\lambda x.f(x)$.

The "class of all cycles" means a collection of all things that have the "cycle" property. It is most likely being called a class because the collection is too large to be contained within a set.

The notation $k \mapsto f(k)$ means that an object $k$ is mapped to the image $f(k)$. This doesn't tell much, but sometimes you could see, for example, (with $x$ a real number):

$$x \mapsto x^2$$

Meaning $x$ would be mapped to its square.

As to the second part, I am not 100% sure but I think they are just referring to all cycles. "The class of all cycles" is just the set with all the cycles in it.

• @JMoravitz thanks ;) – RGS Dec 2 '16 at 1:06