title: What is canonical cover, closure and extraneous attribute?
body: <p>I'm studying database concepts and there are 3 concepts that I don't understand: canonical cover, extraneous attribute and closure. I read the definition about canonical cover but I don't get the picture of how it relates to 3NF and BCNF. The definition of canonical cover appears to be that there are no extraneous attributes and extraneous attributes are attributes that don't change the closure of the set of functional dependencies and closure is the set of all functional dependencies implied by F, a set of functional dependencies.</p>
  
  <p>But all this is a little fuzzy and I'd like to know both an intuitive definition and how to calculate</p>
  
  <ul>
  <li>Canonical cover</li>
  <li>Closure</li>
  <li>Extraneous attribute</li>
  </ul>
  
  <p>Functional dependency I believe I understand--it's like what would have been the PK in a table if we had those attributes in a table. </p>
  
  <p>There is a rather extensive answer at <a href="https://stackoverflow.com/questions/6008354/database-refinement-minimal-cover-of-f-extraneous-attributes">database refinement - minimal cover of F (extraneous attributes)</a> but I found it difficult to read all the set definitions and algebra and I'd rather have definitions in plain English. </p>
  
  <p>For example, having the schema U={A,B,C,D,E,F,G} and the functional dependencies</p>
  
  <pre><code>AB → C
  B → E
  CF → D
  C → A
  B → F
  CE → F
  CD → B
  B → C
  </code></pre>
  
  <p>Are the closures A+,B+,C+,D+,E+,F+ calculated this way?</p>
  
  <pre><code>A+ = A
  B+ = BCDEF
  C+ = A
  D+ = D
  E+ = E
  F+ = F
  </code></pre>
  
  <p>If I'm not mistaken then BCDEFG is a superkey (”the whole key”) in 1NF/2NF but is it minimal (3NF)?</p>
  
  <p>What else should be done to normalize this example to 1NF, 2NF and 3NF with the help of closures and canonical covers? Is canonical cover the same as minimal cover?</p>