20. [10] Find the volume of the set of points $(x, y, z)$ satisfying
$$
\begin{array}{l}
x, y, z \geq 0 \\
x+y \leq 1 \\
y+z \leq 1 \\
z+x \leq 1
\end{array}
$$
side of the plane $x=y$. For each value of $c$ from 0 to $\frac{1}{2}$, the region of the intersection of this half of the solid with the plane $y=c$ is a trapezoid. The trapezoid has height $1-2 c$ and average base $\frac{1}{2}$, so it has an area of $\frac{1}{2}-c$.
The total volume of this region is $\frac{1}{2}$ times the average area of the trapezoids, which is $\frac{1}{2} \cdot \frac{1}{4}=\frac{1}{8}$. Double that to get the total volume, which is $\frac{1}{4}$.