Let A = { 1,2}, and suppose A has the partial order relation where R = { (1,1 ),( 1,2 ), (2,2)

Let S be the set of all strings in a 1's and 2 's and let be the corresponding lexicographic order on S .

Let s and t be any strings in S of lengths m and n, where m and n are positive integers and let s_m and t_m be the characters in the mth position for s and t respectively.

Indicate which of the following statements are true, and for each true statement cite one of the following conditions for your answer:

Condition 1) If m n and the first m characters of s and t are the same, then s t

Condition 2) If the first m - 1 characters in s and t are the same, s_m R t_m, and s_m=t_m, then s t.

Condition 3) If is the null string then s.

* If no strings are related by other than by these three conditions, then is a partial order relation on S.

a) 112 1112 True or False Reason 1, 2, or 3

b) 221 2212 Trueor False Reason 1, 2, or 3

c) 121 Trueor False Reason 1, 2, or 3

d) 1221212 True or False Reason 1, 2, or 3