Step 1: Recall income elasticity.
Income elasticity measures how demand reacts to income,
\[ E_Y=\frac{\%\,\Delta Q}{\%\,\Delta Y} \]
Step 2: Test statement I.
A normal good is bought more as income rises, so $E_Y>0$. Statement I is correct.
Step 3: Test statement II.
An inferior good is bought less as income rises, so $E_Y<0$. Statement II is correct.
Step 4: Test statement III.
A luxury good has elasticity above one, $E_Y>1$. The statement says less than one, so statement III is wrong.
Step 5: Conclude.
So I and II hold while III does not.
\[ \boxed{\text{Both I and II are CORRECT, but III is NOT CORRECT}} \]