Math 368

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Author:
sainy
ID:
15210
Filename:
Math 368
Updated:
2010-04-19 17:03:22
Tags:
Axioms properties
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Description:
axioms and properties of integers
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  1. A1
    • Addition is associative:
    • (x+y)+z = x+(y+z) for all x,y,z elements of Z
  2. A2
    • Addition is commutative:
    • 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

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