# Math 368

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1. A1
• (x+y)+z = x+(y+z) for all x,y,z elements of Z
2. A2
• x+y = y+x for all x,y elements of Z
3. A3
• Z has an identity with respect to addition
• (0)
4. A4
• Every integer x in Z has an inverse with respect to addition...
• (-x)
5. A5
• Multiplication is associative
• (x*y)*z = x*(y*z) for all x,y,z elements of Z
6. A6
• Multiplication is commutative
• x*y = y*x for all x,y elements of Z
7. A7
• Z has a identity with respect to multiplication
• (1)
8. A8
• For all integers x,y,z, x*(y+z) = x*y + x*z
• (distributive laws)
9. P1
If a+b = a+c,then b=c
10. P2
a0 = 0a = 0
11. P3
(-a)b = a(-b) = -(ab)
12. P4
-(-a) = a
13. P5
(-a)(-b) = ab
14. P6
a(b-c) = ab - ac
15. P7
(-1)a = -a
16. P8
(-1)(-1) = 1
17. A9
• Closure Property
• Z+ is closed in Z wrt + and *
• If x,y elements of Z+, then x+y is an element of Z+ and xy is an element of Z+
18. A10
• Trichotomy Law
• For every integer x, exactly one of the following statements is true:
• x is an element of Z+
• -x is an element of Z+
• or x = 0
19. Q1
• Exactly one of the following holds:
• a<b
• b<a
• or a=b
20. Q2
If a>0, then -a<0 and if a<0, then -a>0
21. Q3
If a>0 and b>0,then a+b>0 and ab>0
22. Q4
If a>0 and b<0, then ab<0
23. Q5
If a<0 and b<0, then ab>0
24. Q6
If a<b and b<c, then a<c
25. Q7
If a<b, then a+c < b+c
26. Q8
If a<b and c>0, then ac<bc
27. Q9
If a<b and c<0, then ac>bc
28. A11
• The Well-Ordering Principle
• Every nonempty subset of Z+ has a smalest element; that is, if S is a nonempty subset of Z+, then there exists "a" element of S such that a < x for all x elements of S
 Author: sainy ID: 15210 Card Set: Math 368 Updated: 2010-04-19 21:03:22 Tags: Axioms properties Folders: Description: axioms and properties of integers Show Answers: