Doubt about the logic defining transitivity of a relation Kindly, correct me where I am wrong.
Modified(11:50 AM, 26 March 20): [ for all a, b, c ∈ X ]
Let us define P: $(a, b), (b, c)∈R$ ; and Q: $(a, c)∈R$;
[A] When P is true:


*

*P is true, Q is true: Relation is TRANSITIVE. 

*P is true, Q is false: Relation is NOT TRANSITIVE. 


[B] When P is false
Now, I am going to deal with the condition P is false in this way.
Consider the first case, (P is true and Q is true) $\implies$ Relation is TRANSITIVE.
Then, the negation of this will be (P is false or Q is false) $\implies$ Relation is NOT TRANSITIVE.
This implies that the condition P is false is enough to say that the relation is NOT TRANSITIVE.
Is my argument valid?
 A: No, your argument is wrong because in your analysis of the notion of transitivity you omitted quantifiers and you destroyed the conditional (if... then) structure of the definition.
A binary relation $R$ (over a set $A$) is transitive when, for every $a, b, c \in A$, one has that $(a,b) \in R$ and $(b,c) \in R$ imply $(a,c) \in R$.
What is the negation of the condition above? We have to negate the universal quantifier and the implication!
So, the relation $R$ is not transitive when there exist $a,b,c \in A$ such that $(a,b) \in R, \, (b,c) \in R$ and $(a,c) \notin R$. 
Therefore, to prove that $R$ is transitive you have to take all possible $a,b,c\in A$ (not necessarily distinct) and check that if $(a,b) \in R$ and $(b,c) \in R$ then $(a,c) \in R$. 
Note that in the particular case where  $(a,b) \notin R$ or $(b,c) \notin R$ the implication "if $(a,b) \in R$ and $(b,c) \in R$ then $(a,c) \in R$" is vacuously true.
But you have to check that the implication "if $(a,b) \in R$ and $(b,c) \in R$ then $(a,c) \in R$" holds for all $a, b, c \in A$ to conclude that $R$ is transitive.

Some errors in your analysis are the following: 


*

*Since you omitted quantifiers, you say that, given some $a, b, c \in A$, if you have that $(a,b) \in R$ and $(b,c) \in R$ and $(a,c) \in R$ then $R$ is transitive (case 1 in your analysis). 
This is wrong, because you have shown this just for some $a,b,c \in A$, not for all $a, b, c \in A$.

*Concerning your analysis in the case B, you misunderstand the meaning of the implication. 
In propositional logic, if you are in a situation where $P$ and $Q$ are both true, then the implication $P \to Q$ is true but it does not mean that in all the other situations the implication $P \to Q$ is false. For instance, $P \to Q$ is true also when $P$ and $Q$ are both false.
