Why is an affine set convex?

Question

I wanted to know why do we say that an affine set is convex? From what I understood, if we take two points $x_1$ and $x_2$ $\in \mathbb{R}$, then, the affine set $A$ defined by these two points will be the line that goes through them and the convex set $C$ defined by these two points will be the line segment formed from connecting these two points. So how can an affine set $A$ be convex if we can find points that are $\in A$ but $\not \in C$?

Answer 1

What you are calling the "convex set defined by these points" is usually called the "convex hull", and it is the smallest convex set that contains those points. There are many other convex sets that contain those points though, including the affine line.

"convex" is a property that means if two points $a$ and $b$ are in the set, then the line segment connecting them is also in the set.

Answer 2

A set $X$ is affine if $$ \forall x,y\in X\;\forall t\in\color{red}{\mathbb R}\colon tx+(1-t)y\in X $$ and convex if $$ \forall x,y\in X\;\forall t\in\color{red}{[0,1]}\colon tx+(1-t)y\in X $$ Thus loosely speaking for an affine set we need more tests not more points. Hence affine implies convex but not vice versa.