10A Week 10 - CMPUT 272(Stewart Lecture 17 Reading Epp 1.3...

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CMPUT 272 (Stewart) Lecture 17 Reading: Epp 1.3, 8.1-8.3 Recall: A relation from a set A to a set B is a subset of A × B . If the ordered pair ( a, b ) is in the relation R , we say that a is related to b and we write: ( x, y ) R or xRy . Relations on two or more sets: An n -ary relation on A 1 ×· · ·× A n is a subset of A 1 ×· · ·× A n where A 1 , . . . , A n are sets and n 2. Relations on two sets: A relation from a set A to a set B is a binary relation on A × B . A binary relation on A × A is called a (binary) relation on A . 1
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Examples: 1. R from { 1 , 2 , 3 , 4 , 5 } to { 1 , 2 , 3 , 4 , 5 } ( R on { 1 , 2 , 3 , 4 , 5 } ) where R = { (1 , 1) , (1 , 2) , (1 , 3) , (3 , 4) , (2 , 4) , (4 , 5) , (5 , 4) } . 2. LE from R to R ( LE on R ) where x LE y x y LE = { ( x, y ) R 2 | x y } 3. P from Z to Z ( P on Z ) where x P y x and y are both even or both odd P = { ( x, y ) Z 2 | x and y are both even or both odd } 4. T from P ( { a, b, c } ) to Z where A T x ⇔ | A | ≥ x { 1 } T 0 ( , 2) / T 5. All functions are binary relations but not all binary relations are func- tions. For example, the binary relations above are not functions. 6. B on R 3 where ( x, y, z ) B y < x < z or z < x < y 7. relational databases Let A 1 = the set of all possible student names A 2 = the set of all possible student numbers A 3 = the set of all possible courses A 4 = { A+, A, A-, B+, B, B-, C+, C, C-, D+, D, F } Define R on A 1 × A 2 × A 3 × A 4 as follows: ( w, x, y, z ) R the student with name w and student number x took course y and received a grade of z in that course. 2
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Let R be a relation from A to B . The inverse relation of R is the relation R - 1 from B to A defined as follows: R - 1 = { ( y, x ) B × A | ( x, y ) R } .
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