The "Movies.com" database consists of three relations: MOVIES(MID Category, EIDR), MOVIEVIEWERS(VID, AGE, ZIPCODE), and SUBSCRIPTIONS (MID VID DURATION). Underlined attributes represent keys for the respective relations. Relation SUBSCRIPTIONS has a key that consists of two attributes, MID and VID. Attributes MID and VID in SUBSCRIPTIONS are also foreign keys,w.r.t attribute MID in relation to MOVIES and attribute VID in relation to MOVIEVIEWERS. Attribute "DURATION" in relation to SUBSCRIPTION is always a strictly positive integer. The same applies to the attribute "age" in relation to MOVIEVIEWERS and the attribute "EIDR" in relation to MOVIES (We assume that the EIDR is composed of 4 integers).

 The tables below show a plausible instance of the database:

 

With respect to the database schema presented above, provide an example of two distinct relational algebra expressions that are logically equivalent but syntactically different. Explain in your own words what "query logical equivalence" means.

 

MOVIES MID category |EIDR MOVIEOX01 SCIFI i MOVIE0X02 i SCIENCE |MOVIE0X03 IART I MOVIE0X04 I HISTORY I 0022 i 1314 | 1345 | 6909 MOVIEVIEWERS vid I |-| Joe | 18 Jay | 21 Ali 16 Bob | 29 Zoe | 34 Age SUBSCRIPTIONS ZipCode | | vid | -||-----|- mid | DURATION | Joe | MOVIE0X03 | MOVIE0X01 | | 48126 | | JAY 48127 | 48195 Bob | MOVIE0X01 | 48172 48125 || Joe | MOVIE0X04 ALI MOVIEOX02 25 31 55 25 55