  Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A =
\{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid
a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. We
prove the following results.
  1) The union of two arithmetic progressions (with the same common difference)
is not sum-dominant. This result partially proves a conjecture proposed by the
author in a previous paper; that is, the union of any two arbitrary arithmetic
progressions is not sum-dominant.
  2) Hegarty proved that a sum-dominant set must have at least $8$ elements
with computers' help. The author of the current paper provided a
human-verifiable proof that a sum-dominant set must have at least $7$ elements.
A natural question is about the largest cardinality of sum-dominant subsets of
an arithmetic progression. Fix $n\ge 16$. Let $N$ be the cardinality of the
largest sum-dominant subset(s) of $\{0,1,\ldots,n-1\}$ that contain(s) $0$ and
$n-1$. Then $n-7\le N\le n-4$; that is, from an arithmetic progression of
length $n\ge 16$, we need to discard at least $4$ and at most $7$ elements (in
a clever way) to have the largest sum-dominant set(s).
  3) Let $R\in \mathbb{N}$ have the property that for all $r\ge R$,
$\{1,2,\ldots,r\}$ can be partitioned into $3$ sum-dominant subsets, while
$\{1,2,\ldots,R-1\}$ cannot. Then $24\le R\le 145$. This result answers a
question by the author et al. in another paper on whether we can find a
stricter upper bound for $R$.
