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Commutative laws

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Slideshow:

Communtative (Relational ALgebra) operators
 
The following relational algebre operators are communitative:
Union ∪ Intersection ∩ Cartesian product × Join ⋈
Union (∪) is commutative for sets and bags

Intersection (∩) is commutative for sets and bags

Join (⋈) is commutative

Wait..., how are these sets equal ???
Join (⋈) is commutative
 
If we identify the values according to their position, these sets are different
Join (⋈) is commutative
 
If we identify the values according to their attribute name, these sets are identical !!
Join (⋈) is commutative
 
If we identify the values according to their attribute name, these sets are identical:

Cartesian product (×) is commutative

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• Cartesian product (×) is commutative
  • Cartesian product laws:

    R ×Set S = S ×Set R R ×Bag S = S ×Bag R

    Examples:

    Set A = {1,2} Set B = {3,4} A × B = (a,b) // First attr is from A, second attr is from B B × A = (b,a) // First attr is from B, second attr is from A A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)} B × A = {3,4} × {1,2} = {(1,3), (1,4), (2,3), (2,4)} // Note: after re-ordering to (a,b) !!!

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• Union (∪) is commutative
  • Union laws:

    R ∪Set S = S ∪Set R R ∪Bag S = S ∪Bag R

    Examples:

    Set A = {1,3} Set B = {3,4} A ∪Bag B = {1,3} ∪ {3,4} = {1,3,3,4} B ∪Bag A = {3,4} ∪ {1,3} = {3,4,1,3}

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• Intersection (∩) is commutative
  • Intersection laws:

    R ∩Set S = S ∩Set R R ∩Bag S = S ∩Bag R

    Examples:

    Set A = {1,3,3} Set B = {3,3,4} A ∩Bag B = {1,3,3} ∩ {3,3,4} = {3,3} B ∩Bag A = {3,3,4} ∩ {1,3,3} = {3,3}

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• Join (⋈) is commutative
  • Join laws:

    R ⋈Set S = S ⋈Set R R ⋈Bag S = S ⋈Bag R

    Examples:

    Set A = {(a,1), (a,2), (b,3)} Set B = {(a,5), (b,7)} A ⋈ B = {(a,1), (a,2), (b,3)} ⋈ {(a,5), (b,7)} = {(a,1,a,5), (a,2,a,5), (b,3,b,7)} B ⋈ A = {(a,5), (b,7)} ⋈ {(a,1), (a,2), (b,3)} = {(a,5,a,1), (a,5,a,2), (b,7,b,3)}

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  • Comment:

    Technically speaking, the sets: A ⋈ B = {(a,1), (a,2), (b,3)} ⋈ {(a,5), (b,7)} = {(a,1,a,5), (a,2,a,5), (b,3,b,7)} B ⋈ A = {(a,5), (b,7)} ⋈ {(a,1), (a,2), (b,3)} = {(a,5,a,1), (a,5,a,2), (b,7,b,3)} are different because the position of the values are different However: Each attribute in the join result has a name: A ⋈ B = ( A.attr1 , A.attr2, B.attr1, B.attr2 ) B ⋈ A = ( B.attr1 , B.attr2, A.attr1, A.attr2 ) We refer to the attributes by their names in queries !!! When the attributes in tuples in A ⋈ B and B ⋈ A are re-arranged in the same attribute order: A ⋈ B and B ⋈ A is the same set of tuples !!! Example: Let's re-arrange the result as: ( A.attr1 , B.attr1, A.attr2, B.attr2 ) A ⋈ B = {(a,1), (a,2), (b,3)} ⋈ {(a,5), (b,7)} = {(a,1,a,5), (a,2,a,5), (b,3,b,7)} = {(a,a,1,5), (a,a,2,5), (b,b,3,7)} B ⋈ A = {(a,5), (b,7)} ⋈ {(a,1), (a,2), (b,3)} = {(a,5,a,1), (a,5,a,2), (b,7,b,3)} = {(a,a,1,5), (a,a,2,5), (b,b,3,7)}