Start with every candidate $\mathrm{X}$:
$\pi_{\mathrm{X}}(\mathrm{R})$
Now construct all combinations that each candidate would need in order to qualify:
$\pi_{\mathrm{X}}(\mathrm{R}) \times \mathrm{S}$
This contains every possible required pair:
$(\mathrm{x},\mathrm{y})$
Subtract the pairs that actually exist in $\mathrm{R}$:
$(\pi_{\mathrm{X}}(\mathrm{R}) \times \mathrm{S})-\mathrm{R}$
The result contains the missing required pairs.
Project $\mathrm{X}$:
$\pi_{\mathrm{X}}\left((\pi_{\mathrm{X}}(\mathrm{R}) \times \mathrm{S})-\mathrm{R}\right)$
These are exactly the $\mathrm{X}$ values that fail the "for every" requirement.
So subtract these offenders from all candidates:
$\pi_{\mathrm{X}}(\mathrm{R})-\pi_{\mathrm{X}}\left((\pi_{\mathrm{X}}(\mathrm{R}) \times \mathrm{S})-\mathrm{R}\right)$
Thus,
$\mathrm{R}\div\mathrm{S}=\pi_{\mathrm{X}}(\mathrm{R})-\pi_{\mathrm{X}}\left((\pi_{\mathrm{X}}(\mathrm{R}) \times \mathrm{S})-\mathrm{R}\right)$
Hence C is correct.
The useful interpretation is:
Division = all candidates - candidates missing at least one required pair.
