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\begin{center}
\vskip 1cm{\LARGE\bf Full Subsets of $\mathbb N$}
\vskip 1cm \large {Lila Naranjani}\\
Islamic Azad University\\ 
Dolatabad Branch\\ 
Isfahan\\
Iran\\
\href{mailto:lnaranjani@yahoo.com}{\tt lnaranjani@yahoo.com}  \\
\ \\
Madjid Mirzavaziri\\
Department of Pure Mathematics\\
Ferdowsi University of Mashhad\\
P. O. Box 1159--91775\\ 
Iran\\
\href{mailto:mirzavaziri@gmail.com}{\tt mirzavaziri@gmail.com} \\
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\begin{abstract}
Let $A$ be a subset of $\mathbb N$. We say that $A$ is $m$-full if
$\sum A=[m]$ for a positive integer $m$, where $\sum A$ is the set of
all positive integers which are a sum of distinct elements of $A$ and
$[m]=\{1,\ldots,m\}$. In this paper, we show that a set
$A=\{a_1,\ldots,a_k\}$ with $a_1<\cdots<a_k$ is full if and only if
$a_1=1$ and $a_i\leq a_1+\cdots+a_{i-1}+1$ for each $i, 2\leq i\leq k$.
We also prove that for each positive integer $m\notin\{2,4,5,8,9\}$
there is an $m$-full set. We determine the numbers
$\alpha(m)=\min\{|A|: \sum A=[m]\}, \beta(m)=\max\{|A|: \sum A=[m]\},
L(m)=\min\{\max A: \sum A=[m]\}$ and $U(m)=\max\{\max A: \sum A=[m]\}$
in terms of $m$. We also give a formula for $F(m)$, the number of
$m$-full sets.
\end{abstract}

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