Absolute Value inequality Equivalence relation

Question

For a,b ∈ R, check if $|a−b|\leq5$ is an equivalence relation.

• Reflexive: a ~ a because $|a-a|=0\leq5$
• Symmetric: Let $b~a$, $|b-a|≤5 \Rightarrow |a-b|≤5$, which means that a~b
• Transitive: a~b and $b~c \Rightarrow |a-b|≤5 \Rightarrow -5≤ a-b ≤5$ (1) $|b-c|≤5 \Rightarrow -5≤ b-c ≤5$ (2)

If I add (1),(2) i'll have: $-10≤ a-c ≤10 \Rightarrow |a-c|≤10$ Are the steps correct? Is the relation transitive or no?

Answer 1

The first two steps are correct.

And, no, it is not transitive. In order to prove it, just provide an example. For instance: $0\sim4$ and $4\sim8$, but $0\not\sim8$.