Sums of finitely many distinct rationals

$\mathcal{E}$ denotes the family of all finite nonempty $S\subseteq{\mathbb N}:={1,2,\ldots}$,and${\cal E}(X):={\cal E}\cap{S:S\subseteq X}$when$X\subseteq{\mathbb N}$\mathrm{.}Similarly,${\cal F}$ denotes the family of all finite nonempty $T\subseteq {\mathbb{Q}}^{+}$, and ${\cal F}(Y) := {\cal F}\cap{T:T\subseteq Y}$where${\mathbb Q}^+$ is the set of all positive rationals and $Y\subseteq {\mathbb{Q}}^{+}$. This paper treats the functions $\sigma :\mathcal{E}\to {\mathbb{Q}}^{+}$ given by $\sigma :S\mapsto \sigma S:=\sum \{1\mathrm{/}x:x\in S\}$, the function $\delta :\mathcal{E}\to \mathbb{N}$ defined by $\sigma S=\nu S\mathrm{/}\delta S$ where the integers $\nu S$ and $\delta S$ are coprime, and the more general function $\mathrm{\Sigma }:\mathcal{F}\to {\mathbb{Q}}^{+}$ where $\mathrm{\Sigma }T$ denotes the sum of the elements in $T$ for $T\in \mathcal{F}$. Theorem 1.1. For each $r\in {\mathbb{Q}}^{+}$, there exists an infinite pairwise disjoint subfamily ${\mathcal{H}}_r\subseteq \mathcal{E}$ such that $r=\sigma S$ for all $S\in {\mathcal{H}}_r$. Theorem 1.2. Let $X$ be a pairwise coprime set of positive integers. Then $\sigma$ restricted to $\mathcal{E}(X)$ and $\delta$ restricted to $\mathcal{E}(X)$ are injective. Also, $\sigma C\in \mathbb{N}$ for $C\in \mathcal{E}(X)$ only if $C=\{1\}$. Theorem 6.5. There is a set $X$ of positive rational numbers for which $\mathrm{\Sigma }:\mathcal{F}(X)\to {\mathbb{Q}}^{+}$ is a surjection, but for which $1\in X$ and the only $S\in \mathcal{F}(X)$ with $\mathrm{\Sigma }S=1$ is $S=\{1\}$.