Why $R_3 = \lbrace (1,2),(3,4)\rbrace$ is transitive?
It's like, transitive is said because there's $\{a,b\}$,$\{b,c\}$ then there will be $\{a,c\}$ right? But then, why is that one is said to be transitive?
2026-04-11 18:34:08.1775932448
Determining why this is transitive
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A transitive relation is one such that if we have $(a, b), (b, c) \in R$, then necessarily $(a, c)\in R$.
When we don't have $(a, b), (b, c) \in R$, then we $(a, c)$ need not be in a transitive relation $R$.
Put differently, a relation is transitive unless there exist $(a, b) , (b, c) \in R$, but $(a, c) \notin R$.
In your case, the relation is transitive because there are no pairs of the form $(a, b), (b, c) \in R_3$, so it cannot fail to be transitive. It is vacuously true that the relation is transitive.