Proving Convexity review question

Question

Prove that the set $X=[1,2] \cup [3,4] \subset R $ is not convex.

I know I have to apply the equation $x̄ = \lambda x +(1-\lambda)x'$ where $\lambda$ is in $[0,1]$ as this proves convexity but I don't understand how to go about it and how to prove every point exists in the set $X$.

Answer 1

Short answer: Let $x=2$, $x'=3$, and $\lambda=\frac{1}{2}$. What is $\bar x$? Is it in $X$?

Long answer: Consider the definition.

A set $X\subseteq\mathbb{R}$ is convex if for all $x$ and $x'$ in $X$ and $\lambda$ in $[0,1]$, the combination $\bar x = \lambda x + (1-\lambda)x'$ is in $X$.

To negate this contradiction, you need to invert the quantifiers:

A set $X$ is not convex if there exist $x$ and $x'$ in $X$ and $\lambda \in [0,1]$ such that $\bar x = \lambda x + (1-\lambda)x'$ is not in $X$.

What the definition says in English is that “the line between any two points of $X$ is contained in $X$.” So to show that a set is not convex, you need to find two points of $X$ such that the connecting line between them is not contained in $X$. That led me to pick $x=2$ and $x'=3$. Letting $\lambda = \frac{1}{2}$ just makes $\bar x$ the midpoint $2\frac{1}{2}$, obviously not in $X$.