Very Simple Relations

Question

For Set A = {1,2,3,4}

Is it possible to generate a relation that is reflexive and symmetric, but not transitive? The textbook says, (1,1),(2,2),(3,3),(4,4),(1,2),(2,1)(2,3)(3,2) but isn't this transitive?

Answer 1

No it is not transitive, because it contains $(1,2)$ and $(2,3)$ but not $(1,3)$.

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An example on the real numbers: If $x\sim y$ means $|x-y|<1$, then $\sim$ is reflexive, symmetric, and not transitive.

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The example you gave could be modified slightly by removing $4$ from $A$ (and $(4,4)$ from the relation), while still being reflexive, symmetric, and not transitive. In that case, it could be defined as $a\sim b$ if and only if $|a-b|\leq 1$.

Answer 2

You have $1\sim 2$ and $2\sim 3$; do you also have $1\sim 3$, as required for transitivity? (I’m using $\sim$ to denote the relation.)

Answer 3

It is not transitive since $(1,2)$ and $(2,3)$ are in $R$ but $(1,3)$ is not.