26-RelsOpsProps

26-RelsOpsProps - Discussion #26 Relations: Operations...

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Discussion #26 Chapter 5, Sections 3.4-4.5 1/15 Discussion #26 Relations: Operations & Properties
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Discussion #26 Chapter 5, Sections 3.4-4.5 2/15 Topics Inverse (converse) Set operations Extensions and restrictions Compositions Properties reflexive, irreflexive symmetric, antisymmetric, asymmetric transitive
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Discussion #26 Chapter 5, Sections 3.4-4.5 3/15 Inverse If R: A B, then the inverse of R is R ~ : B A. R -1 is also a common notation R ~ is defined by {( y , x ) | ( x , y ) R}. If R = {(a,b), (a,c)}, then R ~ = {(b,a), (c,a)} > is the inverse of < on the reals Note that R ~ ~R The complement of < is But the inverse of < is > (sometimes called converse)
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Discussion #26 Chapter 5, Sections 3.4-4.5 4/15 Set Operations Since relations are sets, set operations apply (just like relational algebra). The arity must be the same indeed, the sets for the domain space and range space must be the same (just like in relational algebra).
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Discussion #26 Chapter 5, Sections 3.4-4.5 5/15 Restriction decreases the domain space or range space. Example: Let < be the relation {(1,2), (1,3), (2,3)}. The restriction of the domain space to {1} restricts < to {(1,2), (1,3)}. Extensions increase the domain space or range space. Example: Let < be the relation {(1,2), (1,3), (2,3)}. The extension of both the domain space and range space to {1,2,3,4} extends < to {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)}
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26-RelsOpsProps - Discussion #26 Relations: Operations...

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