What is the relationship between an affine space and an affine set?

Question

I just learned what an affine space is and its basic properties. It is formally defined as $\Bbb A=(A,V,f)$ where $A$ is a set of points, $V$ is a vector space, and $f:A^2→V$ is a function st. for any $x,y,z∈A$, we have

1. $f(x,y)+f(y,z)=f(x,z)$;
2. $f_o (x):A→V≔f(o,x)$ is a bijection from $A$ to $V$ for any fixed $o∈A$.

Here $f(x,y)$ is also denoted as $y-x$.

I also learned that an affine set is defined as (using field $\Bbb R$ for example) the set $\{\theta x+(1-\theta)y:\theta\in\Bbb R\}$ where $x,y\in\Bbb R^n$ are fixed. This concept is introduced in the first lecture of a mathematical optimization course.

I am a bit confused now about what is the relationship between the two concepts, since they are both titled "affine"? Thank you.