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We are using following definition of 3NF:

A schema R is in third normal form (3NF) if for all FD α → β in F +, at least one of the following holds:

  1. α → β is trivial (i.e., β ⊆ α).
  2. α is a superkey for R
  3. Each attribute A in β – α is contained in a candidate key for R (prime).

I don't understand the third condition for this definition. OK, each atribute A in β –β – α means? What set of attributes does it include? α is contained in candidate key of R. What set of attributes does it include?

philipxy
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Marc
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  • It means a(alpha) is a prime attribute, since any part of a candidate key means its a prime attribute. – Rinkesh P May 17 '22 at 10:30
  • This is unclear/strange. Why "β –β – α", it seems a typo for "β – α". What do you mean, "α is contained in candidate key of R"? α → β is a FD in F+. A is an attribute that may be in a CK. How are you parsing (3)? Where are you 1st stuck? Where are you 1st unable to identify a symbol or expression? Otherwise you're expecting us to do that for you with no idea what you misunderstand or don't or do understand. PS When giving a definition use iff/when, not if. Using if relies on the reader to use the fact that it's a definition to properly misinterpret the if as an iff/when. PS Other typos too. – philipxy May 17 '22 at 22:46

1 Answers1

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β – α is the set of the attributes β minus the attributes that are in α, if any. So the third rule says that we can have attributes determinated by something which is not a superkey, but only if those attributes are primes (i.e. part of a candidate key), and obviously are not already in the determinant α (otherwise we have a trivial dependency).

So, when a relation schema is not in third normal form? When all the three conditions are false: that is when we have at least a non-trivial dependency whose determinant is not a superkey (so neither a candidate key) and whose right part contains attributes that are not prime.

Renzo
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