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Venn Diagrams
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  1. Venn Diagrams Database Principles

  2. Venn Diagram • Venn Diagrams are used to represent relationships between sets. • They can also be used to represent set operations like union, intersection and set difference. • Since Relational Algebra has operations corresponding to these set operations, Venn Diagrams are a useful design tool for Relational Algebra Queries. union intersection set difference Database Principles

  3. Exercise: • Draw a Venn Diagram showing the following three sets: • The set of all suppliers • The set of suppliers of red parts • The set of suppliers of non-red parts • Label each part of the diagram. All Suppliers Suppliers who supplynothing at all Suppliers of Red Parts Suppliers of both Red and non-Red Parts Suppliers of non-Red Parts Database Principles

  4. Database Principles Query Types: • Hard Queries: These are queries that have words such as “only”, “all” and “no” in the query condition. • Easy Queries: Queries without such words in the condition. • What makes a query “easy” is that is can be answered with at most join, select and project operators. • What makes a query “hard” is that you must use set difference or quotient to answer the query. Find the suppliers of all/no/only red parts Find the suppliers who supply some red part

  5. Database Principles Easy Queries: • Venn Diagrams do not play much part in solving easy queries. The problem is too easy to need a design phase. • Easy queries are solved by joining all necessary tables, selecting the rows of interest and projecting the columns in the answer.

  6. Database Principles Hard Queries: • In probability, it is often easier to calculate the probability of the complement of an event (¬A) than the probability of the event itself (A). • This same approach can be used to solve hard queries. • Asked to find something hard: • Start by stating and finding its complement • Then use the set difference operator to throw away the newly found complement • You are left with what you want Remember Pr(A) = 1 – Pr(¬A)

  7. “Only” Query • Find the suppliers (SName) who supply only red parts. PartSuppliers = πSNo(Supplies) . s5 . s2 . s3 . s1 suppliers of only red parts = ? . s4 Describe this set. OnlyRedPartSuppliers = PartSuppliers \ NonRedpartSuppliers FinalAnswer = πSName(Supplier OnlyRedPartSuppliers) suppliers of at least onepart that is not red NonRedParts = πPNo(σColour != ‘red’ (Part)) NonRedPartSuppliers =πSNo(Supplies NonRedParts) Database Principles

  8. Database Principles Exercise: • In the previous Venn Diagram, find out where the Suppliers who supply nothing at all are located. • It turns out they are not in the answer set. • What if we want them in the answer set? Replace with PartSuppliers = πSNo(Supplies) PartSuppliers = πSNo(Supplier)

  9. Negation Queries: • Entity Properties: • Location of a Supplier is a property modeled as an attribute • What Parts a Supplier supplies is also a property modeled as a relationship Database Principles

  10. Negation Queries (Some are Easy): • Find the Suppliers who do not come from Boston. Find all entity instances where the value of Location is not Boston Query Mechanism: Look at all the entity instances and as you find one where the value of Location is something other than ‘Boston’, pick out and return that instance as part of the answer. SuppliersNotFromBoston = σLocation != ‘Boston’ (Supplier) Database Principles

  11. Negation Queries (Some are Hard): • Find the Suppliers who do not supply any Parts. Find all entity instances that do notparticipate in the <supplies> relationship. Query Mechanism: Pick a Supplier. Look at every row in the Supplies tableand if you fail to see that Supplier even once then consider that Supplier as part of the answer set. Then move on to a new Supplier. So instead of selecting members of the answer set looking at rows one at a time(which is what happens when we use join) we need to consider all the rows in Supplies as a set and see if a particular Supplier is completely missing or not. NonSuppliers = πSNo(Supplier) \πSNo(Supplies) Database Principles

  12. Negation Queries (Summary): • Negation queries that negate the value of an attribute are “easy” and can be resolved using join, select & project. • Negation queries that negate the participation in a relationship are “hard” and need set difference. Database Principles

  13. Database Principles Quotient Queries: • Queries that contain “all” or “every” in the query condition. • Venn diagrams not very useful. Q = R/S where:Q = the key to the thing you are looking forS = the key to the thing described in the “all” condition R = a table that contains the above two keys and is related to the query. S = the list of key values that satisfy the “all” condition Example: Find the books reserved by all cardholders from New PaltzQ = {isbn}, since we are looking for booksS = {borrowerid}, since borrowerid is the key for cardholders S = πborrowerid(σb_addr = ‘New Paltz’ (Cardholder)) R = {isbn,borrowerid}, must combine Q and S. R = πisbn,borrowerid(Reserves) Q = R/S = {the books reserved by all cardholders from New paltz}