Prove that, If $S$ and $T$ are convex sets, $S \cap T$ is a convex set.
As I understand it, a convex set is a set such that, for any two points, $A$ and $B$, from the set, the line, $AB$, lies within the set.
The proposition is that, If $S$ and $T$ are convex sets, $S \cap T$ is a convex set; however, this does not seem correct to me. I do not see how the intersection of two sets, although independently convex, results in another single set that is necessarily convex?
My Workings
A (hypothesis): $S$ and $T$ are convex sets.
B (conclusion): $S \cap T$ is a convex set.
B1: Given two points $A$ and $B$ from the set $S \cap T$, the line $AB$ lies within the set.
B2: For all points $A$ and $B$ from the set $S \cap T$, the line $AB$ lies within the set.
B2 rephrases B1 using the universal quantifier.
A1: Let $A \in S$ and $B \in T$ such that $A \in S \cap T$ and $B \in S \cap T$.
A2: … ?
At this point, it is not clear to me that, for all points $A$ and $B$ from the set $S \cap T$, the line $AB$ lies within the set; therefore, I have struggled to continue formulating a proof.
I would greatly appreciate it if someone could please take the time to explain what I am misunderstanding and how to correctly continue with my proof.
Best Answer
You are on the right track. Note that $A\in S\cap T$ and $B\in S\cap T$ implies that $A\in S$ and $B\in S$. Since $S$ is convex, line $AB$ lies within $S$. By similar argument line $AB$ lies within $T$. Hence line $AB$ lies within $S\cap T$.