General Computer Science; Problems and Solutions for ... - Kwarc

1.1.4 Relations and Functions
Problem 1.8 (Associativity of Relation Composition)
Let R, S, and T be relations on a set M. Prove or refute that the composition operation for
relations is associative, i. e. that
Solution:
Proof:
P.1 Let 〈x, y〉 ∈ ((T ◦ S) ◦ R).
P.2 ∃z1 ∈ M.〈x, z1〉 ∈ R ∧ 〈z1, y〉 ∈ (T ◦ S)
(T ◦ S) ◦ R = T ◦ (S ◦ R)
P.3 ∃z1, z2 ∈ M.〈x, z1〉 ∈ R ∧ (〈z1, z2〉 ∈ S ∧ 〈z2, y〉 ∈ T )
P.4 ∃z2 ∈ M.〈x, z2〉 ∈ (S ◦ R) ∧ 〈z2, y〉 ∈ T
P.5 〈x, y〉 ∈ (T ◦ (S ◦ R)).
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