Why isn't $\{(1,1),(1,2),(2,1)\}$ transitive relation?

Question

As we knows that Transitive relation R is (a, b) € R and (b, c) € R implies (a, c) € R

So we can suppose that let a be 1 and b be 2 and c be 1 Then, (a, b) € R means (1,2) € R and (b, c) € R means (2,1) € R implies (a, c) € R means (1,1) € R

Then it must be transitive .... But why isn't? Plz plz tell me fastly

Answer 1

In order to be a transitive relation on some set X, R has to satisfy: $$a\text{R}b \quad \text{and} \quad b\text{R}c \quad \Rightarrow \quad a\text{R}c, \quad \text{for each} \quad a,b,c \in X.$$

In your example, we can take $a=2, b=1, c= 2$, and it is obvious that $a$ is in relation with $b$ and that $b$ is in relation with $c$. But, it is also obvious that $a$ is not in relation with $c$, so your relation is not transitive.

Answer 2

The relation isn't transitive because while $(2, 1), (1, 2)$ are in the relation, $(2, 2)$ is not.