Intersection of Relations
The intersection of two relations contains those ordered pairs that appear in both relations. This creates a new relation that is the 'common part' of the two relations.
Formal Definition
In the intersection, only those connections remain that are present in both relations.
Examples of Intersection of Relations
Let R = { (1,2), (2,3), (3,4) }, S = { (2,3), (3,4), (4,5) }.
Then R ∩ S = { (2,3), (3,4) }.
Properties
- The intersection is always narrower or equal to the original relations.
- The intersection is commutative: R ∩ S = S ∩ R.
- The intersection is associative: (R ∩ S) ∩ T = R ∩ (S ∩ T).
- When examining properties of two relations, the intersection often inherits them (e.g., if R and S are transitive, then R ∩ S is also transitive).
Summary
The intersection of relations contains those connections that are common to both. This is useful when we want to highlight the common part of two different connections, for example in network, family, or transportation system analysis.
Practice Exercise
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