Thread: relations transitive

How come 2) is transitive and not 1)

1) R_1 = (1,1) (1,2) (2,1)
2) R_2 = (2,1) (3,1) (3,2) (4,1) (4,2) (4,3)

Definition of transitive relation states that for every(a,b,c) belongs to R
(a,b)belongs to R AND (b,a)belongs to R then (a,c) belongs to R.

So how come 1) is NOT transitive and 2) IS transitive?

Can anyone clarify my doubt!?

Thank you.

Hello, robocop_911!

$\displaystyle 1)\;\;R_1 \:= \:\{(1,1),\:(1,2),\:(2,1)\}$
$\displaystyle 2)\;\; R_2 \:=\:\{ (2,1),\: (3,1),\: (3,2),\: (4,1),\: (4,2),\: (4,3)\}$

Definition of transitive relation states that:
for every $\displaystyle a,b,c \in R,\;(a,b) \in R\,\text{ AND }\,(b,c) \in R$, then $\displaystyle (a,c) \in R.$

So how come 1) is NOT transitive and 2) IS transitive?

In $\displaystyle R_1$, we have:

$\displaystyle (1,1) \in R_1 \:\wedge \:(1,2) \in R_1\quad\Rightarrow\quad (1,2) \in R_1$ . . . True

$\displaystyle (1,2) \in R_1 \:\wedge \:(2,1) \in R_1 \quad\Rightarrow\quad (1,1) \in R_1$ . . . True

$\displaystyle (2,1) \in R_1 \;\wedge \;(1,2) \in R_1 \quad\Rightarrow\quad (2,2) \in R_1$ . . . False!

. . $\displaystyle R_1$ is not transitive.



In $\displaystyle R_2$, we have:

$\displaystyle (4,2) \in R_2 \:\wedge \:(2,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2$ . . . True

$\displaystyle (4,3) \in R_2 \:\wedge \:(3,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2$ . . . True

. . $\displaystyle R_2$ is transitive.

Originally Posted by Soroban

Hello, robocop_911!


In $\displaystyle R_1$, we have:

$\displaystyle (1,1) \in R_1 \:\wedge \1,2) \in R_1\quad\Rightarrow\quad (1,2) \in R_1$ . . . True

$\displaystyle (1,2) \in R_1 \:\wedge \2,1) \in R_1 \quad\Rightarrow\quad (1,1) \in R_1$ . . . True

$\displaystyle (2,1) \in R_1 \;\wedge \;(1,2) \in R_1 \quad\Rightarrow\quad (2,2) \in R_1$ . . . False!

. . $\displaystyle R_1$ is not transitive.



In $\displaystyle R_2$, we have:

$\displaystyle (4,2) \in R_2 \:\wedge \2,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2$ . . . True

$\displaystyle (4,3) \in R_2 \:\wedge \3,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2$ . . . True

. . $\displaystyle R_2$ is transitive.

I disagree...
$\displaystyle (2,1) \in R_1 \;\wedge \;(1,2) \in R_1 \quad\Rightarrow\quad (2,2) \in R_1$ . . . False!

. . $\displaystyle R_1$ is not transitive.

it could bye (2,1) ^ (1,1) => (2,1) and (2,1) is in the set!

Recall that transitive property says for all elements a,b,c a~b, b~c => a~c

Originally Posted by Isomorphism

Recall that transitive property says for all elements a,b,c a~b, b~c => a~c

I am considering All the elements! Still, I am not getting the correct answer.

How come only...

R_6 = (3,4) is Also transitive? This is really mindboggling!