Okay, so I have a homework problem I'm a little confused about,

The textbook is pretty useless and we didn't go into types of orders very much in class. So, am I to show that the dictionary order is reflexive, antisymmetric, and transitive on XxX, since XxX is already linearly ordered? I hadn't even heard of the dictionary order until I saw this problem, so I'm a little confused as to how to start it off.

I've usually heard it called "lexicographic order". Anyways...

No it's not! XxX is just a set!

Your goal is to show [itex](X \times X, \preceq)[/itex] is a total order...

which means you have to do this.

Just plow forward and do it. There's no trick to it, no cleverness is required: you just brute force your way through the logic. You know you're supposed to prove this ordering to be reflexive, antisymmetric, and transitive. So, just start trying to prove it reflexive! What does it mean for this ordering to be reflexive?