If $A=\{1, 2, 3\}$ and $R=\{(1, 1),(2, 2), (1, 2), (2, 1), (1, 3)\}$ then is $R$ transitive?

Question

OK this is a general question. How do I determine if a relation given to me is transitive or not?

Let the following be sets $A=\{1, 2, 3\}$ and $R=\{(1, 1),(2, 2), (1, 2), (2, 1), (1, 3)\}$ then is $R$ transitive or not and why? Please explain with more examples.

Answer 1

Transitivity of a relation $R$ is the statement that $(a,b) \in R \wedge (b,c) \in R \implies (a,c) \in R$, where $\wedge$ is the "and" symbol and $\implies$ is the "implies" symbol, in terms of logical representation. This is sometimes simplified to the notation $aRb \wedge bRc \implies aRc.$ In words, this essentially means that given element $a$ and $b$ are related in this relations and $b$ and $c$ are related in this relation, then $a$ and $c$ are related in this relation. The way you would determine if a relation is transitive or not is to show that if $(a,b),(b,c) \in R,$ $(a,c) \in R.$ As an example with our current $R$,

$$(1,1),(1,1) \in R \implies (1,1) \in R$$ $$(1,1),(1,2) \in R \implies (1,2) \in R$$ $$(1,1),(1,3) \in R \implies (1,3) \in R$$ $$(2,2),(2,1) \in R \implies (2,1) \in R$$ $$(2,2),(2,2) \in R \implies (2,2) \in R$$ $$(2,1),(1,1) \in R \implies (2,1) \in R.$$

The last several transitivity requirements are not as trivially implied. Since $(1,2),(2,1) \in R$, for $R$ to be transitive, it needs to be that $(1,1) \in R$; Thankfully, this holds in $R$. Similarly, $(2,1),(1,2) \in R$ must also have $(2,2) \in R$ hold for $R$ to be transitive; This is satisfied as well in $R$.

However, $(2,1),(1,3) \in R,$ but $(2,3) \notin R$. Thus, transitivity does not hold in $R$.

Answer 2

R is a transitive relation if for every $a,b,c\in A$, $$(a,b),(b,c)\in R$$then $$ (a,c)\in R$$
When $A=\{1, 2, 3\}$ and $R=\{(1, 1),(2, 2), (1, 2), (2, 1), (1, 3)\}$
You have $(2,1),(1,3)\in R$, but $(2,3)\notin R$, so R is not transitive.