Exercise 275Logic, sets and proofs › RelationsHigh schoolAdded 3 Nov 2026

Transitive relations

#transitive#relation#definition

Statement

Define a transitive relation and test whether $a\mathcal{R}b\iff a-b\in\mathbb{Z}$ on $\mathbb{R}$ is transitive.

Hint

Transitive: $a\mathcal{R}b$ and $b\mathcal{R}c\Rightarrow a\mathcal{R}c$ .

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Definition. $\mathcal{R}$ transitive if $(a\mathcal{R}b\land b\mathcal{R}c)\Rightarrow a\mathcal{R}c$ .

Test. Suppose $a-b\in\mathbb{Z}$ and $b-c\in\mathbb{Z}$ . Then $a-c=(a-b)+(b-c)\in\mathbb{Z}$ . So $a\mathcal{R}c$ .

Conclusion. This relation is transitive (it is $\equiv\pmod1$ ).