Why are two random variables independent if the Pearson's correlation coefficient equals zero, but the same result does not hold for covariance?

I was reading the following book

Han J, Pei J, Kamber M. Data mining: concepts and techniques. Elsevier; 2011 Jun 9. (Third Edition)

On page 96, at the first line of the last paragraph it says (here)

If the resulting value is equal to $0$, then $A$ and $B$ are independent and there is no correlation between them.

where the resulting value above corresponds to the following formula (correlation coefficient)

$$r_{A,B}=\frac{\sum_{i=1}^n (a_i - \overline{A}) (b_i - \overline{B})}{n\sigma_A\sigma_B}. \tag{3.3}$$

However, on the next page on the last paragraph, it says

If $A$ and $B$ are independent (i.e., they do not have correlation), then ... $Cov(A,B) = \ldots = 0$.

Up to here, everything looks good, however by the following relation $$r_{A,B} = \frac{Cov(A,B)}{\sigma_A\sigma_B} \tag{3.5}$$ the correlation and covariance are related and as far as I remember, if the covariance of two random variables tend to be zero, it is not necessary that they are independent. However, the book says if $r_{A,B} = 0$ , then $A$ and $B$ are independent. Am I right that the book is wrong? or there is something else happening here.

• When all the correlations are 0 it is the off diagonal elements that should be 0. Zero correlation implies independence for a bivariate normal but not in general for other distrbutions. – Michael R. Chernick Apr 10 '17 at 19:19
• Did you buy this book because it's required for a class? I can't think of other reasons to buy this thing – Aksakal Apr 10 '17 at 19:19
• If you read just a little further in the book, it explicitly tells you that zero covariance does not imply independence: see the bottom of p. 97. – whuber Apr 10 '17 at 19:38
• I will wholeheartedly agree that the book is not well written. – whuber Apr 10 '17 at 19:46
• Possible duplicate of Under what additional conditions does independence follow from zero correlation? – Dilip Sarwate Apr 11 '17 at 3:03