Consider the contracting similarities S1(x)=x/3, S2(x)=(x+λ)/3, S2(x)=(x+2)/3 where λε[0, 1]. Assuming Fλ is the invariant set with respect to S1, S2, and S3. In this paper, we study the Hausdorff measure, Hausdorff dimension and the structure of F. Letλ= b a 2 Q \ [0, 1], (a, b)=1. By using combinatorial techniques, we prove that |Fλ|>0 if a≡b 6≠0 (mod 3) and that in the other cases Fλ is a Cantor-like fractal with recursive construction.