  Let $K$ be the attractor of the following IFS
  $$\{f_1(x)=\lambda x, f_2(x)=\lambda x +c-\lambda,f_3(x)=\lambda x
+1-\lambda\}, $$
  where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap
f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main results of
this paper are as follows: $$\sqrt{K}+\sqrt{K}=[0,2]$$ if and only if
$$\sqrt{c}+1\geq 2\sqrt{1-\lambda},$$ where
$\sqrt{K}+\sqrt{K}=\{\sqrt{x}+\sqrt{y}:x,y\in K\}$. If $c\geq (1-\lambda)^2$,
then $$\dfrac{K}{K}=\left\{\dfrac{x}{y}:x,y\in K, y\neq
0\right\}=\left[0,\infty\right).$$ As a consequence, we prove that the
following conditions are equivalent:
  (1) For any $u\in [0,1]$, there are some $x,y\in K$ such that $u=x\cdot y;$
  (2) For any $u\in [0,1]$, there are some $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,
x_9,x_{10}\in K$ such that $$u=x_1+x_2=x_3-x_4=x_5\cdot x_6=x_7\div
x_8=\sqrt{x_9}+\sqrt{x_{10}};$$
  (3) $c\geq (1-\lambda)^2$.
