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?
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.