Math Help - 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!

Definition of transitive relation states that:
for every , then

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

In , we have:

. . . True

. . . True

. . . False!

. . is not transitive.



In , we have:

. . . True

. . . True

. . is transitive.

Originally Posted by Soroban

Hello, robocop_911!


In , we have:

1,2) \in R_1\quad\Rightarrow\quad (1,2) \in R_1" alt="(1,1) \in R_1 \:\wedge \1,2) \in R_1\quad\Rightarrow\quad (1,2) \in R_1" /> . . . True

2,1) \in R_1 \quad\Rightarrow\quad (1,1) \in R_1" alt="(1,2) \in R_1 \:\wedge \2,1) \in R_1 \quad\Rightarrow\quad (1,1) \in R_1" /> . . . True

. . . False!

. . is not transitive.



In , we have:

2,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" alt="(4,2) \in R_2 \:\wedge \2,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" /> . . . True

3,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" alt="(4,3) \in R_2 \:\wedge \3,1) \in R_2 \quad\Rightarrow\quad (4,1) \in R_2" /> . . . True

. . is transitive.

I disagree...
. . . False!

. . 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!