Example 2 Given the set $M=\{(x, y)|| x|+| y \mid<1\}$ and $N=\left\{(x, y) \left\lvert\, \sqrt{\left(x-\frac{1}{2}\right)^{2}+\left(y+\frac{1}{2}\right)^{2}}\right.\right.$ $\left.+\sqrt{\left(x+\frac{1}{2}\right)^{2}+\left(y-\frac{1}{2}\right)^{2}}<2 \sqrt{2}\right\}$, find the relationship between sets $M$ and $N$.
Analysis and Solution: We observe its characteristics and use the combination of number and shape to solve the problem. Set $M$ represents the interior of the square $ABCD$ with vertices at $A(0,1)$, $B(-1,0)$, $C(0,-1)$, and $D(1,0)$; Set $N$ represents the interior of the ellipse with foci at $F_{1}\left(-\frac{1}{2}, \frac{1}{2}\right)$, $F_{2}\left(\frac{1}{2},-\frac{1}{2}\right)$ and a major axis of $2 \sqrt{2}$. Since the four vertices of the square $ABCD$ are all inside the ellipse, $M$ is a proper subset of $N$.