$A = \{1 , 2 , 3\}$ and $R = \{(1 , 1) , (2 , 2) \}$. Can anyone please tell me how would I explain that the relation $R$ is transitive?

Question

Let $A$ be a set. $A = \{1 , 2 , 3\}$.

$R = \{(1 , 1) , (2 , 2) \}$. Can anyone please tell me how would I explain that the relation $R$ is transitive ?

My Attempt: I taught my student first what is non-transitive. If we get two elements such that $(a , b) , (b , c)$ but the $a$ is not related to $c$ , then we call the relation Non-transitive. If a relation is not a Non-transitive , we call that a Transitive Relation.

So the given relation is not a non-transitive. Since there is no such case.

Can anyone give me a better way to help 10+2 level students understand what a transitive relation is?

Answer 1

For this relation $R$, we have $a\mathrel Rb$ when and only when $a=b$ and $a,b\in\{1,2\}$. So, if $a\mathrel Rb$ and $b\mathrel Rc$, we have that $a=b$, that $b=c$, and that $a,b,c\in\{1,2\}$. So, $a=c$ and $a,c\in\{1,2\}$, which means that $a\mathrel Rc$.

Of course, by the same argument, if $A$ is any set and if $R\subset\{(a,a)\mid a\in A\}$, then $R$ is transitive.

Answer 2

Let $(a,b), (b,c) \in R.$ Since $A$ consists consists of two elements, therefore either $c=a$ or $c=b.$

Case 1: $c=a$ Then you want to show $(a,a)\in R$ which is true.

Case 2: $c=b$ Then you want to show $(a,b) \in R$ which is true.

Answer 3

Definition of transitivity of a relation $\rho$ on a set $S$ is, if $a\rho b$ and $b\rho c$ then $a\rho c$ for any $a,b,c\in S$.

Now, here $\rho =\{(1,1),(2,2)\}$. So for the assumption $a\rho b$ and $b\rho c$, $b=1$ or $2$.

If $b=1$ then $a=c=1$, which gives $a\rho c$.

Next, If $b=2$ then $a=c=2$, which also gives $a\rho c$.

Hence the condition gets satisfied. Thus, the relation $\rho$ is transitive.

Answer 4

The answer lies in the truth value of conditional statement $p\implies q$.

A conditional $p\implies q$ is true (T) whenever the hypothesis $p$ is false (F) irrespective of the truth value of $q$. Mathematicians call it vacuous truth. The definition of transitive i.e.

If $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$.

is a conditional $p\implies q$ where

$p$: $(a,b)\in R, (b,c)\in R$

$q$: $(a,c)\in R$

So whenever you find that the hypothesis $p$ is false, the relation $R$ becomes transitive.