Table 4: Do the properties of a population resulting from a certain pairing function depend on its generating properties? The three generating properties fM(M), fq(q) and $\mathcal{B}$ are listed in the top row. For each of the ten quantities listed in the left-most column, we list whether or not they depend on the choice of fM(M), fq(q) and $\mathcal{B}$. We indicate dependence and independence with the symbols $\surd$ and $\times $, respectively. The results for fq;M1(q) and $\mathcal{B}_{M_1}(M_1)$ are valid only for a population of binaries with identical primary mass M1. The distribution over total system mass $f_{M_{\rm T}}$ includes both binary systems and single stars.
Generating prop. fM(M) fq(q) $\mathcal{B}$
Pairing function \rotatebox{-90}{\mbox{$\!\!\!\!$ RP}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCRP}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-I}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-II}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-III}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-I}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-II}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-III}} \rotatebox{-90}{\mbox{$\!\!\!\!$ RP}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCRP}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-I}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-II}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-III}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-I}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-II}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-III}} \rotatebox{-90}{\mbox{$\!\!\!\!$ RP}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCRP}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-I}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-II}} \rotatebox{-90}{\mbox{$\!\!\!\!$ PCP-III}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-I}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-II}} \rotatebox{-90}{\mbox{$\!\!\!\!$ SCP-III}}
fM1 $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ - - $\times $ $\surd$ $\times $ $\surd$ $\surd$ $\surd$ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $
fM2 $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ - - $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $
$f_{M_{\rm T}}$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ - - $\surd$ $\surd$ $\surd$ $\times $ $\surd$ $\times $ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\times $ $\surd$ $\times $
fM1,S $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ - - $\times $ $\times $ $\times $ $\surd$ $\surd$ $\surd$ $\surd$ $\times $ $\times $ $\times $ $\times $ $\surd$ $\surd$ $\surd$
$f_{M_{\rm S}}$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ - - $\times $ $\surd$ $\times $ $\times $ $\surd$ $\times $ $\times $ $\times $ $\times $ $\surd$ $\times $ $\times $ $\surd$ $\times $
$f_{M_{\rm all}}$ (``IMF'') $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ - - $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\times $ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
$f_{q,{\rm all}}(q)$ $\surd$ $\surd$ $\times $ $\surd$ $\surd$ $\times $ $\surd$ $\surd$ - - $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $
fq;M1(q) $\surd$ $\surd$ $\times $ $\times $ $\times $ $\surd$ $\surd$ $\surd$ - - $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $ $\times $
$\mathcal{B}_{\rm all}$ $\times $ $\times $ $\times $ $\surd$ $\times $ $\times $ $\surd$ $\times $ - - $\times $ $\surd$ $\times $ $\times $ $\surd$ $\times $ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
$\mathcal{B}_{M_1}(M_1)$ $\surd$ $\times $ $\times $ $\times $ $\times $ $\surd$ $\surd$ $\surd$ - - $\times $ $\surd$ $\times $ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$


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