Lexicographic order and order on pairs

Statement

Is the alphabetical order ($\le_{\text{lex}}$) on words total or partial? And the order on pairs of real numbers with the rule $(a,b)\le(c,d)$ if and only if $a\le c$ and $b\le d$?

Solution

Alphabetical order: it is total. Given two distinct words, they are compared letter by letter up to the first difference (or to the end of the shorter one): the outcome always establishes which comes first. Every pair is therefore comparable.

Order on pairs (product order): it is partial. For example $(1,2)$ and $(2,1)$ are not comparable, because $1\le 2$ but $2\not\le 1$: neither of the two pairs is “less than or equal to” the other.

Links

Topics: Set theory
Concepts: Partial order · Total order
Skills: Proving
Exercise type: Proof