Lexicographical order in $\Bbb R^n$?

Question

Consider the set $ \Bbb R^n = \{ x = (x_1, ..., x_n) : x_1, ..., x_n \in \Bbb R \}.$

For $x,y\in \Bbb R^n$, we define $<$,$\leq$ as below: $$ x<y \iff j \in \{1,..,n \} (x_j<y_j) \wedge \forall i \in \Bbb N (i<j \to x_i=y_i).$$

$$x \leq y \iff (x<y) \vee (x=y). $$
$"<"$ defines on $ \Bbb R^n $ the so called "Lexicographical Order".

Prove:
$(a)$ $ \leq $ on $ \Bbb R^n $ defines a order-relation.
$(b)$ $( \Bbb R^n, \leq )$ is total ordered.

So, could you give me any hints about it, that let me solve alone this exercise? And what is correlation between the lexicographical order and the $\leq$ realtion? Thank you!

Answer 1

Compare on the first component and in case of a tie compare on the second component and in case of a tie compare on the third component...

$\le$ is the non-strict version of $<$.

Antisymmetry: if two elements compare both ways ($\le$ and $\ge$), then their first components compare both ways, and there is a tie on the first components; and so on with the next components.

Transitivity: if the pairs $(x, y)$ and $(y, z)$ are ordered, then their first components are ordered and by transitivity the first components of $(x, z)$ are ordered. In the case of a tie on $(x, z)$, there is a tie on the first components of $(x, y)$ and $(y, z)$, and the reasoning applies to the next components.

Totality: the first components can be compared and in case of a tie the second components can be compared...

Answer 2

To answer your question about the correlation, suppose you have a dictionary of all the 4 letter words in the English language. For instance, you have the words $x$ and $y$ where $(x_1,x_2,x_3,x_4) = (D,A,N,G)$ and $(y_1,y_2,y_3,y_4) = (D,A,R,N)$. The reason that DANG comes before DARN in the dictionary is that if we take $j=3$ then $x_3=N$ comes before $y_3=R$ in the alphabet, whereas for $i=1,2$ which are the values of $i<j$ we have $x_1=y_1=D$ and $x_2=y_2=A$.

Now if you represent the 26 letters of the alphabet in order as the integers along the real line, namely $1$ for $A$, and $2$ for $B$, etc., then $(D,A,N,G)$ is represented in $\mathbb{R}^4$ as $x=(4,1,14,7)$ and $(D,A,R,N)$ is represented as $y=(4,1,18,14)$, and we see that $x<y$ in the lexicographic order, just as explained above.