Graphs of min(|x|,|y|)=1 , min(|x|,|y|)>=1 , max(|x|,|y|)=1 and max(|x|,|y|)<=1

Question

Hey I have trouble understanding these graphs, I tried plotting x=+-1 and y=+-1 but dont get how the min and max play a part in this, Can anybody explain this to me inuitively thanks. Using Desmos I saw that for min we get 4 L shaped lines and for max we get a box, also will be helpful if you could explain the inequalies as well.

Answer 1

Let's take $\min(|x|,|y|) \ge 1$ and understand which points fit.

Let's do the analysis in the first quadrant, and then you can extend it to the other quadrants by symmetry.

In the first quadrant, $x,y$ are positive so we need to understand where $\min(x,y) \ge 1$ would lie. Notice if both $x,y<1$ this does not hold, so we need all places where either $x\ge 1$ or $y \ge 1$ or both.

So you get the set $\left\{(x,y) \in \mathbb{R}^2 \right| \left. x \ge 1 \text{ or } y \ge 1\right\}$. Can you finish?