Relational Algebra Symbols

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1 Relational Algebra Symbols Unary Operators Selection p (T) takes a subset of rows, p is a predicate like a 1 > 3 Projection (a1, a2,...an) (T) takes a subset of columns, c1 cn are columns Renaming ρ (a1, a2,, an) (E) renames an expression (not on the test) Binary Operators Union T S combines the rows of two tables Intersection T S returns the rows that both tables have in common Difference T S returns all the rows of T that are not in S Cross Product T S returns all possible combinations of the rows of T joined with the rows of S, see example Theta Join T p S equivalent to p (T S) where p is any predicate Equijoin T T.an = S.an S equivalent to p (T S) where p is T.c n = S.c n Natural Join T S equivalent to the Equijoin but without the columns T and S are joined from Not on Test, but good to know when we start doing SQL: Left Semijoin T S equivalent to (T.a1, T.a2,, T.an) (T S) Right Semijoin T S equivalent to (S.a1, S.a2,, S.an) (T S) Left Outer Join T S equivalent to (T S) but includes rows of T that fail to join on S, putting null values for attributes that would have been added from S Right Outer Join T S equivalent to (T S) but includes rows of S that fail to join on T, putting null values for attribute that would have been added from T rows from S T have null values for S columns Full Outer Join T S equivalent to (T S) but includes rows from both

2 S and T that fail to join on each other, putting null values for attributes that would have been added from the other table rows from S T have null values for S columns rows from T S have null values for S columns Division T S see example Aggregation Count, Min, Max, Avg, Sum, see example Grouping works in tandem with aggregation but sub-divides the aggregation by rows whose specified properties have matching values, see example

3 Relations for Examples: NonFictions Title AuthorFirst AuthorLast Godel, Escher, Bach Douglas Hofstadter Bleeding Edge J.D. Kleinke The Art of Deception Kevin Mitnick In Cold Blood Truman Capote On the Road Jack Keroac Authors FirstName LastName Truman Douglas Thomas Capote Hofstadter Pynchon Fictions Title AuthorFirst AuthorLast Gravity s Rainbow Thomas Pynchon In Cold Blood Truman Capote On the Road Jack Keroac Bleeding Edge Thomas Pynchon Bleeding Edge William Johnstone

4 1. Find all fiction books whose title is Bleeding Edge. Title= Bleeding Edge (Fictions) = Bleeding Edge Thomas Pynchon Bleeding Edge William Johnstone 2. Give a list of all nonfiction authors first and last names. (firstname, lastname) (Nonfictions) = Douglas J.D. Kevin Truman Jack Hofstadter Kleinke Mitnick Capote Keroac 3. Give a complete list of fiction and nonfiction books with author names: Fictions Nonfictions = Godel, Escher, Bach Douglas Hofstadter Bleeding Edge J.D. Kleinke The Art of Deception Kevin Mitnick In Cold Blood Truman Capote On the Road Jack Keroac Gravity s Rainbow Thomas Pynchon Bleeding Edge Thomas Pynchon Bleeding Edge William Johnstone 4. Give a list of books with authors that are considered both fiction and nonfiction. Fictions Nonfictions In Cold Blood Truman Capote On the Road Jack Keroac

5 5. Give a list of books with authors that are fiction but are not also considered nonfiction Fictions Nonfictions Gravity s Rainbow Thomas Pynchon Bleeding Edge Thomas Pynchon Bleeding Edge William Johnstone

6 Relations for Joins: MusicArtists BandName Founded NumberOfMembers Bassnectar Lumineers Beyonce Nas Albums AlbumName ReleaseYear BandName The Lumineers 2012 Lumineers 2011 Bassnectar Cleopatra 2016 Lumineers Lemonade 2016 Beyonce 2003 Beyonce

7 Cross Product: MusicArtists X Albums = AlbumName ReleaseYear BandName BandName Founded NumberOfMembers The Lumineers 2012 Lumineers Bassnectar Bassnectar Bassnectar Cleopatra 2016 Lumineers Bassnectar Lemonade 2016 Beyonce Bassnectar Beyonce Bassnectar Bassnectar The Lumineers 2012 Lumineers Lumineers Bassnectar Lumineers Cleopatra 2016 Lumineers Lumineers Lemonade 2016 Beyonce Lumineers Beyonce Lumineers Lumineers The Lumineers 2012 Lumineers Beyonce Bassnectar Beyonce Cleopatra 2016 Lumineers Beyonce Lemonade 2016 Beyonce Beyonce Beyonce Beyonce Beyonce The Lumineers 2012 Lumineers Nas Bassnectar Nas Cleopatra 2016 Lumineers Nas Lemonade 2016 Beyonce Nas Beyonce Nas Nas

8 Theta Join: MusicArtists NumberOfMembers > 1 Albums = AlbumName ReleaseYear BandName BandName Founded NumberOfMembers The Lumineers 2012 Lumineers Lumineers Bassnectar Lumineers Cleopatra 2016 Lumineers Lumineers Lemonade 2016 Beyonce Lumineers Beyonce Lumineers Lumineers Recall that this operation is just a Selection on the cross product of MusicArtists and Albums (consult previous table, which is the cross product of these two tables). Here is the Theta Join rewritten as the selection of the cross product NumberOfMembers > 1 (MusicArtists X Albums) Equijoin: Show all albums paired with their respective artists: MusicArtists MusicArtists.BandName = Albums.BandName Albums = AlbumName ReleaseYear BandName BandName Founded NumberOfMembers 2011 Bassnectar Bassnectar The Lumineers 2012 Lumineers Lumineers Cleopatra 2016 Lumineers Lumineers Lemonade 2016 Beyonce Beyonce Beyonce Beyonce Rewriting the Equijoin as a selection of the cross product we have: MusicArtists.BandName = Albums.BandName (MusicArtists X Albums) It produces the same table above as the Equijoin.

9 Natural Join: The natural join produces the same table as an Equijoin but reduces the doubled column(s) on which the two tables are joined to a single set: MusicArtists Albums = AlbumName ReleaseYear BandName Founded NumberOfMembers 2011 Bassnectar The Lumineers 2012 Lumineers Cleopatra 2016 Lumineers Lemonade 2016 Beyonce Beyonce Notice that the predicate defining the columns on which the join operates is implicit in this notation. Also notice that only one BandName column remains instead of one copy from each of the original tables. There will be another review sheet for the remaining joins, aggregation, and grouping at a later time. They will be covered alongside their SQL counterparts. You are not responsible for the other joins, aggregation, division, and grouping on Test 1 because we haven t adequately covered them yet.

Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules

+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that