Set Theory: Symmetric Relations

I was just trying to figure out this problem I came across. For a set
$X = \{1, 2, 3, 4, 5\}$ is it possible to come up with a relation on $X$ that is symmetric, but neither reflexive nor transitive?

Also, it is possible to find a relation that is transitive, but neither reflexive nor symmetric?

Thank you!

Answer

In what follows I use $aR\,b$ as an abbreviation for $(a,b)\in R$.

To attack problems like these, start with a small relation that satisfies the positive condition. For your second problem, for instance, you could begin by letting $1 R\,2$, $2 R\,3$, and $1R\,3$; if you stop there, you have a transitive relation. Is it reflexive? Is it symmetric? Can you stop at this point?

Of course, you could have started simply with $1R\,1$, which is also transitive, but it’s clearly symmetric, so that’s a bad idea. The same objection rules out starting with $\varnothing$, the empty relation, in which nothing is related to anything: it’s also symmetric. The next simplest way to get transitivity is the one that I actually used.

The same approach works for your first problem and yields the answer suggested by JDH: starting with $1R\,1$ isn’t very helpful, but starting with $1R\,2$ and $2R\,1$ works without further modification.