Question

Suppose R is the relation {(a, b),(c, d)} and S is the relation {(b, c),(d, e)}.

Let R ◦ S be the composition of R and S. True or false:
a) (a, d) ∈ R ◦ S
b) (c, e) ∈ R ◦ S
c) The relation R ◦ S has at least three ordered pairs in it

Answer 1

The statements (a, d) ∈ R ▶ S and that R ▶ S has at least three ordered pairs are false. The statement (c, e) ∈ R ▶ S is true.

Step-by-step explanation:

The composition of two relations R and S, denoted R ▶ S, consists of ordered pairs (x, z) such that there exists a y for which (x, y) is in R and (y, z) is in S.

a) The pair (a, d) is in R ▶ S if there exists a b such that (a, b) is in R and (b, d) is in S. In this case, (a, b) is in R and (b, c) is in S, but since there is no (b, d), (a, d) ∈ R ▶ S is false.

b) The pair (c, e) is in R ▶ S if there exists a d such that (c, d) is in R and (d, e) is in S. In this case, (c, d) is in R and (d, e) is in S, therefore (c, e) ∈ R ▶ S is true.

c) For R ▶ S to have at least three ordered pairs, there would need to be at least three distinct pairs that meet the condition for composition. Given R and S only provide two distinct pathways (a to b to c, c to d to e), there can't be three pairs, thus the statement that R ▶ S has at least three ordered pairs is false.