• Support PF! Buy your school textbooks, materials and every day products Here!

Spivak Calc: Is my proof correct?

  • Thread starter albert1993
  • Start date
  • #1
18
0

Homework Statement


Prologue Chapter 1. Problem 8.

3OEq7.png


(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
 

Answers and Replies

Related Threads on Spivak Calc: Is my proof correct?

  • Last Post
Replies
5
Views
2K
Replies
12
Views
3K
  • Last Post
Replies
1
Views
937
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
1
Views
891
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
1
Views
829
Replies
3
Views
710
  • Last Post
Replies
3
Views
571
Top