If represents the lexicographic order which of the following

If represents the lexicographic order, which of the following is/are true?

(5,12) (5,4)

(5,4) (5,12)

(5,4) and (5,12) are not comparable because we need the first number to be smaller in one of the pairs.

Solution

(a,b) < (c,d) when:
a < c or
a = c and b < d

a) false
b) true
c) false (By above definition)

Alternate:

Given two partially ordered sets A and B, the lexicographical order on the Cartesian product A × B is defined as

(a,b) (a,b) if and only if a < a or (a = a and b b).

a1a2 ... ak

b1b2 ... bk

if and only if at the first i where ai and bi differ, ai comes before bi in the alphabet.

Answer b) (5,4) (5,12)

5=5, then 412 and 4<12