Question

Can somebody help me understand a transitive relation. I understand both reflexive and symmetric relations, but I am having issues understanding transitive relations.

Answer 1

for example let's consider this relation:
\[a|b\], namely a divides b, or a is a divisor of b

Answer 2

which means that exists a integer k such that b = a*k

Answer 3

I can state that \[a|b\]
is an equivalence relation

Answer 4

ok

Answer 5

now we can check the transitive property as follows:
if
\[a|b\quad {\text{and}}\quad b|c\]

Answer 6

then I state that
\[a|c\]

Answer 7

here is the proof:
by definition, we know that exists an integer k such that:
b=k*a
furthermore, by definition, I know that exists an integer h, such that:
c=h*b

Answer 8

so we can write this:
\[c = hb = h\left( {ka} \right) = \left( {hk} \right)a\]

Answer 9

since hk is again an integer number, then we can can say that c divides a, namely the thesis

Answer 10

that is an example of transitive property

Answer 11

so does that mean that there is a relationship that connects every three members?

Answer 12

better is to say an implication, namely:
\[{\text{if }}\left( {a|b\quad {\text{and}}\quad b|c} \right)\quad {\text{then: }}a|c\]

Answer 13

but say for the sake of understanding...I'm trying to visualize it...is it true?

Answer 14

yes!

Answer 15

ok, got it thanks

Answer 16

thanks!