Discrete Math Notes:
Chapter 5: Functions and Relations
5.9 Composition of relations
Composition, is another relation on A, denoted S o R. The pair (a, c) E S o R if and only if there is a b E A such that (a, b) E R and (b, c) E s.
Overview:
Relation R and S on set A ={ a, b, c, d }
SoR
Captions ^
1. R and S are relations on set A (a, c) E S o R because there is a vertex d such that (a,d) E R and (d,c) ES. 2. Since (c,d) E R and (d,c) E S, (c,c) E S o R 3. Since (b,b) E R and (b,d) E S, (b,d) E S o R. 4.Since (b,b) E R and (b,a) E S, (b,a) E S o R.
Another relation on A, labeled S o R, is the composition of relations R and S on set A. If and only if there is a b A such that (a, b) R and (b, c) E S, the pair (a, c) E S o R. Because it is natural to read an expression from left to right, the ordering of the relations R and S in the statement "S o R" may appear strange. The ordering established here for the composition of relations, on the other hand, is consistent with the ordering defined for the composition of functions, in which R is applied first, followed by S.