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Consider the following three relations in a relational database. Employee(eId, Name), Brand(bId, bName), Own(eId, bId) Which of the following relational algebra expressions return the set of eIds who own all the brands? 

Consider the following three relations in a relational database. Employee(eId, Name), Brand(bId, bName), Own(eId, bId) Which of the following relational algebra expressions return the set of eIds who own all the brands?  Correct Answer Π<sub>eld</sub> (<span style="">Π</span><span style=" line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">eld,bld</span><span style="">​ (Own)/Π<sub>b</sub></span><span style=" line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">ld</span><span style="">​ (Brand))</span>, <span style="">Π</span><span style=" line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">eld</span><span style="">​ (Own) - Π</span><span style=" line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">eld</span><span style="">​ ((Π</span><span style=" line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">eld</span><span style="">​ (Own) × Π<sub>b</sub></span><span style=" line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">ld</span><span style="">​ (Brand)) - Π</span><span style=" line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em; font-size:10.5px;">eld,bld</span><span style="">​ (Own))</span>

Concept:

  • The "Division operator" in relational algebra return all the entities that are associated with entities of different relation.

 

The result of the query in option 1:
It will display all the eId of relation "Own" which are associated with all the bId of relation "Brand".

The result of the query in option 2: 

At first, the cartesian product is performed between Own(eId) and Brand(bId). Which results in the association of all eId with all bId. 

Now, subtract the Own relation entities from the newly generated relation which contains the cartesian product. This will result in eIds that are not associated with bIds.

Subtracting these resulted eIds from the Own relation to retrieve the eIds that are associated with bId in the Brand relation, which is similar to the option 1 result.

Πeld (Πeld,bld​ (Own)/Πeld​ (Brand)) = Πeld​ (Own) - Πeld​ ((Πeld​ (Own) × Πeld​ (Brand)) - Πeld,bld​ (Own))

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