So, I have a set $A=\{1,2,3\}$, and I have to do a bunch of different relations for it, but I can't seem to be able to grasp as to what exactly I'm supposed to do here. One of the asked questions is like so:
Form a relation to set $A$ so that the relation is reflexive, but is not symmetrical nor transitive.
The logic I can understand well enough, and I realize that I have to form a relation that fulfills
$xRx$, for $x\in A$
$xRy\not =yRx$ for $x, y \in A$
$xRy \land yRz \not => xRz$ for $x, y, z \in A$
but I can't figure out how I can actually form the relation, apart from writing out the rules of the relation.
The second logical condition you've written down is not quite correct---in fact (since $A$ is nonempty), the special case $y = x$ contradicts the first contradiction. Symmetry means that $x R y$ iff $y R x$ for all $x, y \in A$, and the negation of this condition is that there is some choice of $x, y \in A$ such that $x R y$ but not $y R x$.
The third condition is also not quite correctly written, for basically the same reason that the second is not.
As for finding a relation, one can, as you say, simply construct one from scratch, and there are many choices. I recommend doing this once if you haven't already---the process may be illuminating, even if the result is now.
Here's a more intuitive source for constructing a relation.
Hint Consider a relation that defines the rules of the familiar game rock-paper-scissors.