Question

Digraph relations, True or False

Answer 1

A relation \(R\) on \(X={1,2,3}\) is transitive if \(1R2\), \(2R3\) and \(1R3\)

Answer 2

False.
The definition states that in order for \(R\) to be transitive on \(X\) we must have for all \(x, y, z\in X\) that if \(xRy\) and \(yRz\) then \(xRz\) also, so it is not sufficient to check that \(1R2\), \(2R3\) and \(1R3\).

Answer 3

I understand that the difference is between the `and` and the `then`, but I still don't quite understand how that makes a difference.

Answer 4

@mathmate @wio

Answer 5

uh look at truth tables

Answer 6

a->b is
TT T
TF F
FT T
FT T

Answer 7

a ^ b is
TT T
TF F
FT F
FF F
also for a-> B the last line should be FF T

Answer 8

in other words 1R2^2R3^1R3 is a necessary but NOT sufficient to for R to be transitive

Answer 9

so you are saying that it is possible that the three can be interlinked through different relations, so that means \(1R2\), \(2R3\) and \(1R3\) but not because of each other?