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I came across following definition of 2NF normal form:

A relation schema is in second normal form (2NF) if every nonprime attribute is not partially dependent on any key of R.

Then the book reworded the definition as follows:

A relation schema R is in 2NF if every nonprime attribute A in R is fully functionally dependent on every key of R.

However I don't feel that these two definitions are equivalent. In first definition "every nonprime attribute is not partially dependent on any key of R" means there can be a nonprime attribute which might be partially dependent on other non prime attribute, but not at all dependent (either partially or fully functionally) on any key. This is not allowed by second definition as it says "every nonprime attribute is fully functionally dependent on every key". Is the second definition wrong?

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