20 - with updating and maintaining integrity Consider the...

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We have shown that ( R S ) T R ( S T ) . The reverse direction showing that R ( S T ) ( R S ) T can be proved in a similar way. P ROPOSITION 3.10 Let R and S be arbitrary binary relations on A . In general 1. R = R - 1 ; 2. composition is not commutative: that is, R S = S R ; 3. R R - 1 = id A . Proof Just as for proposition 2.7, the way to prove that a property does not hold is to provide a counter-example. A counter-example to part 1 is the relation R = { ( a, b ) } { a, b } × { a, b } . Then R - 1 = { ( b, a ) } which is plainly different from R . To show that composition is not commutative, we must find R, S such that R S = S R . Let A = B = { a, b } , R = { ( a, a ) } and S = { ( a, b ) } . Then R S = { ( a, b ) } but S R = . Part 3 is left as an exercise. 3.4 Application to Relational Databases A relational database is a collection of relations. We describe further oper- ations on relations which are key operations used in relational databases.
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Unformatted text preview: with updating and maintaining integrity.] Consider the example of a uni-versity registry database, which has a relation Student storing the students’ names, addresses and examination numbers. It is usual to represent such a database relation as a table: name address number . . . . . . . . . Brown, B 5 Lawn Rd. 105 Jackson, B. 1 Oak Dr. 167 Smith, J. 9 Elm St. 156 Walker, S. 4 Ash Gr. 189 . . . . . . . . . Each tuple of the relation corresponds to a row in the table. The records in a database, in this case name , address , number , are called the attributes of the relation; each attribute corresponds to a column. Associated with each 21...
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