I just started Analysis and I have a few questions,

**1.** "In a partial order on $\displaystyle X$, an element $\displaystyle x \in X$ is

*maximal* if $\displaystyle (y \in X) \wedge (x \leq y) \Rightarrow y = x$, and $\displaystyle x$ is

*maximum* if $\displaystyle z \leq x$ for all $\displaystyle z \in X$"

It seems to me that these definitions are equivalent, since if $\displaystyle (y \in X) \wedge (x \leq y) \Rightarrow y = x$ it means that $\displaystyle \nexists\ y \in X$ such that $\displaystyle x < y$, so x is bigger than every other element in the set.

Also, $\displaystyle \forall z \in X,\ z \leq x$ is pretty much saying x is bigger than every other element in the set, right?

So what am I missing? :/

In the definition of maximal there is no requirement that if $\displaystyle y\in X\,\,\,then\,\,\,y\leq x$ , and in the definition of maximum THERE IS such a requirement. For example, if you look at the set of sets $\displaystyle S:=\left\{\{1,2,3\}\,,\,\{2\}\,,\,\{1,3,4,5\}\righ t\}$ and you partial order it wrt subset relation, then both $\displaystyle \{1,2,3\}\,,\,\{1,3,4,5\}$ are maximal elements of S since there is no set (element) in S "bigger" (containing) any of them. Now, if you'd put $\displaystyle S'=S\cup \{1,2,3,4,5\}$ then $\displaystyle \{1,2,3,4,5\}$ is now the maximum element of S'. Pay attention to the fact that is an element is a maximum then it also is a maximal element and, in fact, the only one (maximum/maximal) there is, but there can be many maximal elements. Finally, wrt your last question: that an element is maximal means there is no element bigger than it, and that an element is a (in fact, as already noted, THE) maximum means it is the biggest element of the set. **2.**
"A subset Y of X such that for any $\displaystyle x,y \in Y$, either $\displaystyle x \leq y$ or $\displaystyle y \leq x$ is called a chain. If X itself is a chain, the partial order is a linear or total order."

Just to make sure I'm understanding this alright:

Say you have a set $\displaystyle X=\lbrace\lbrace 1,2\rbrace,\lbrace 3\rbrace,4,5,6\rbrace$

Then would $\displaystyle Y = \lbrace \lbrace 1,2\rbrace, \lbrace 3\rbrace \rbrace$ with partial order $\displaystyle \subseteq$ and $\displaystyle Z = \lbrace 4,5,6\rbrace$ with partial order $\displaystyle \leq$ be "chains"?

Y is not a chain because neither $\displaystyle \{1,2\}\subset \{3\}$ nor $\displaystyle \{3\}\subset \{1,2\}$ . Z is a chain because it contains one unique element so it vacuously fulfills the condition.
What are other partial orders you could have? I can only think of $\displaystyle \leq, \geq, \subseteq, \supseteq$

There are infinitely many different partial orders on infinitely many different sets, but perhaps it's too soon to even give you examples. First grab the basics. **3.** Do partially ordered sets have to be countable?

No **4.** What does $\displaystyle \mathbb{R}^\mathbb{N}$ mean?

Per definition it is the set of all the functions from the natural numbers to the real ones. You can think of it as the set of all the real sequences... Tonio
Thanks