# Trying to Understand the Meaning of Transitivity in Relation to a Particular Problem

I was trying to understand transitive relation and so I was solving a problem. The question is : $$R_1 = \{(a,b)| a =b \text{ or }a = -b\} , R_2 = \{(a,b)| a =b \}, R_3 = \{(a,b)| a =b+1\}$$, which one is transitive and why?

As far as I know transitive relation is, if $$a>b$$ and if $$b > c$$, then $$a> c$$. I am assuming that the given a and b are real numbers but I am not sure where I will get C so that I can show that which one is transitive.

I saw many youtube tutorials and read my book but I am very confused with this math. I am new in this topic. It will be really helpful if someone can please explain how can I solve this problem.

Thank you very much.

Relations are just collections of ordered pairs. For example, a relation on the set of real numbers $$\mathbb{R}$$ would be $$(x,y)$$ pairs. Formally, a relation on a set $$S$$ is simply some subset of the cartesian product $$R \subseteq S \times S$$. We say a relation $$R$$ is transitive if whenever the pairs $$(a,b), (b,c) \in R$$ then $$(a,c) \in R$$ as well. In the first relation, if we know $$(a,b), (b,c) \in R_1$$, then we know either $$a = b$$ or $$a = -b$$. Similarly, either $$b = c$$ or $$b = -c$$. Clearly this implies either $$a = c$$ or $$a = -c$$ as we know the magnitudes of a,b,c must be the same. Thus $$(a,c) \in R_1$$ and the relation is transitive.
$$R_2$$ is the identity relation. Or simply, the equality $$=$$ is transitive. That is, if $$(a,b)$$ and $$(b,c)$$ are the both in $$R_2$$, then $$a=b$$ and $$b=c$$. Thus $$a=c$$, i.e. $$(a,c)\in R_2$$.
Try replacing, for example, your $$a \gt b$$ with $$\{a,b\} \in R_1$$, $$b \gt c$$ with $$\{b,c\} \in R_1$$ and $$a \gt c$$ with $$\{a,c\} \in R_1$$. Then replace $$R_1$$ with $$R_2$$ and $$R_3$$ to see for which of these the first $$2$$ statements (e.g., if $$\{a,b\} \in R_1$$ and $$\{b,c\} \in R_1$$) means the third one must hold as well (e.g., $$\{a,c\} \in R_1$$).
If you do this, you should find that $$R_1$$ and $$R_2$$ are transitive, while $$R_3$$ is not (because $$a = b + 1$$ and $$b = c + 1$$ means $$a = c + 2$$, not $$a = c + 1$$). Can you finish the rest yourself?