# Understanding transitive relations on set $\{0,1,2,3\}$

I'm having a hard time understanding the transitive property for the following relation. I believed it to be transitive and I can't determine why it is not:

Example 1: $$\{(0,0),(1,1),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\}$$

Case 1: $$((0,0)\in R \wedge (0,0) \in R) \to (0,0) \in R$$

Case 2: $$((1,1)∈R∧(1,1)∈R)→(1,1)∈R$$

Case 3: $$((1,3)∈R∧(3,1)∈R)→(1,1)∈R$$

Case 4: $$((2,2)∈R∧(2,2)∈R)→(2,2)∈R$$

Case 5: $$((2,3)∈R∧(3,2)∈R)→(2,2)∈R$$

Case 6: $$((3,1)∈R∧(1,3)∈R)→(3,3)∈R$$

Case 7: $$((3,2)∈R∧(2,3)∈R)→(3,3)∈R$$

Case 8: $$((3,3)∈R∧(3,3)∈R)→(3,3)∈R$$

Since $$∀_a ∀_b ∀_c (((a,b)∈R∧(b,c)∈R)→(a,c)∈R)$$ is true for all cases, is it not the case that this is a transitive relation?

• Just as with your prior question all you need is one counterexample. Here we have $2\sim_R 3$ and $3\sim_R 1$ but we do not have $2\sim_R 1$.
– lulu
Commented Apr 7, 2019 at 14:49
• This tutorial explains how to typeset mathematics on this site. Commented Apr 7, 2019 at 14:56

You did not check all cases. It is not transitive because both $$(1,3)$$ and $$(3,2)$$ belong to it, whereas $$(1,2)$$ doesn't.