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albert1993
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Homework Statement
Prologue Chapter 1. Problem 8.
(P10) Trichotomy law: For every number a, one and only one of the following holds:
(i) a = 0
(ii) a is in the collection P
(iii) -a is in the collection P
(P11) If a and b are in P, then a + b are in P
(P12) If a and b are in P, then a x b is in P.
The Attempt at a Solution
P10: Let (b-a) be any number d. Then P'10 reduces to
i) d = 0
ii) 0 <d
iii) 0 >d
This is in essence what P10 is, except with the variable being different
P11: Let there be numbers a, b, c, d.
If a<b and c<d, then by P'12 the following are true:
1) a+c < b+c
2) b+c < b+d
By P'11, a + c < b + d
Looking at the case where 0 < b and 0 < d, b + d > 0 which is P11.
P12: If 0 < a and 0 < c, by P'13, 0 < ac, which is P12