Let $$A$,$ be a totally ordered set, and consider the Cartesian product A $\backslash$times A$.$ The total order $$ on A induces a relation on A $\backslash$times A called the lexicographic order, denoted $$lex$,$ which is defined as follows: For all $$a$,$ b$,$c$,$ d$$ in A $\backslash$times A$,$ we say that

$$a$,$ b$$lex $$c$,$ d$$ if and only if one of the following conditions holds:

$$a$,$b$=$c$,$d$,$ or

$$ a$$canda$=$c$,$or $$ a$=$candb$$d$.$

Notice that this is similar to the way words are arranged in a dictionary; hence the name $$lexicographic$.$

Although we are calling the relation $$lex the lexicographic $$order$$ on A $\backslash$times A$,$ we have not justified using the term $$order$.$ In fact, it turns out that whenever $$ is a total order on a set A$,$ the induced relation $$lex is a total order on the set A $\backslash$times A$.$ Further, if $$ is a well order on A$,$ then the induced relation $$lex is well order on A $\backslash$times A$.$ We will prove these facts in the next two questions.

$($a$)$ Show that the relation $$lex is a total order on A $\backslash$times A$.$