Q36: Case study: one-way traffic flow between five spots as a relation

A relation is a set of ordered pairs. Here the pair $(x, y)$ is in R exactly when the data say traffic flows from $x$ to $y$⁠. The order matters, because the flow is one-way.

R is reflexive only if $(a, a) \in R$ for every element $a$ of the set. So one missing pair is enough to show it is not reflexive.

None of the listed flows starts and ends at the same spot, so $(A, A)$ is not in R.

R is transitive if, whenever $(a, b) \in R$ and $(b, c) \in R$⁠, also $(a, c) \in R$⁠. To show it is not transitive, one counterexample is enough.

Traffic goes A to B and B to E, but the data list no flow from A to E.

Watch out: some chains do work, for example $(A, B)$⁠, $(B, C)$ and $(A, C)$⁠. That does not make R transitive; the rule must hold for every chain.

Each sentence of the data gives ordered pairs (from, to): 3 pairs from A, 2 from B, 1 from C and 2 from D. That is $3 + 2 + 1 + 2 = 8$ pairs.

The domain of a relation is the set of first elements of its pairs, and the range is the set of second elements.

First elements, in order: A, A, A, B, B, C, D, D. Without repeats: $\{A, B, C, D\}$⁠.

Second elements, in order: B, C, D, C, E, E, E, C. Without repeats: $\{B, C, D, E\}$⁠.

No traffic leaves E, so E is not in the domain. No traffic arrives at A, so A is not in the range.

A relation is a function only if every element of the domain has exactly one image. A has three images (B, C and D), so this fails.

The same is true of B and D, which have two images each.