This is a question my son Bob asked me. For some sets it is relatively easy to test for membership but a lot more difficult to find members, and for others the reverse is true. Here is an elementary example to get the idea across. An $m \times n$ real matrix $M$ defines a linear map $x \mapsto M x = y$, from ${\mathbb R}^n$ to ${\mathbb R}^m$. It is easy to test if $x$ is in the kernel; just compute $M x$ and see if it is zero, but to find an $x$ in the kernel you must solve $M x = 0$ which is more computationally intensive. Conversely it is easy to find an element in the range; just choose any $x$ and compute $M x$; but to test if $y$ is in the range you must solve $M x = y$. Does anyone know if there is a standard name for this distinction or for sets of these two types?