Composition of relations

Definition

The composition of relations is an operation that combines two binary relations to form a new relation. If we have two relations, R and S, the composition of R and S, denoted as R ∘ S, consists of all pairs (a, c) such that there exists an element b making (a, b) in R and (b, c) in S. This concept is crucial in understanding how different relations interact and can be used to derive new relations from existing ones.

5 Must Know Facts For Your Next Test

1. The composition of two relations can be visualized as 'connecting' the elements of the first relation to the elements of the second through an intermediary.
2. For two relations R and S defined on sets A and B, and B and C respectively, the resulting composition R ∘ S is a relation from A to C.
3. If R is a transitive relation and S is equal to R, then the composition R ∘ R will also be transitive.
4. The composition operation is not commutative; that is, R ∘ S is not necessarily equal to S ∘ R.
5. The composition of relations can be used in various fields such as computer science, mathematics, and logic to describe complex relationships.

Review Questions

• How does the composition of relations relate to the concept of transitivity?
  • The composition of relations is closely linked to the concept of transitivity. If you have a transitive relation R, and you take its composition with itself (R ∘ R), the result will also be transitive. This means that if you can go from element a to b through R and from b to c through R again, then you can directly relate a to c through the composed relation. Understanding this connection helps in analyzing how relations build on each other.
• Discuss the implications of the non-commutativity of relation composition using examples.
  • The non-commutativity of relation composition means that for two relations R and S, the results of R ∘ S and S ∘ R may not be the same. For example, if R represents 'is a parent of' and S represents 'is a teacher of', then while one can compose these to say 'a parent can also be a teacher', reversing this doesn't hold true in all cases. This property emphasizes the directional nature of relational operations and is essential for understanding complex relationships in systems.
• Evaluate how the composition of relations can be applied in computer science, particularly in database management.
  • In computer science, particularly within database management systems, the composition of relations is vital for querying data across multiple tables. When querying data using joins, you are effectively composing relations where one table's foreign key connects to another's primary key. This allows for complex data retrieval where relationships between various data sets are established. Analyzing how these compositions operate helps in optimizing queries and understanding data integrity across relational databases.

Related terms

binary relation: A binary relation on a set is a collection of ordered pairs of elements from that set.

transitive relation: A relation is transitive if whenever it relates a to b and b to c, it also relates a to c.

inverse relation: The inverse of a relation R consists of all pairs (b, a) such that (a, b) is in R.