Math 368
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A1
 Addition is associative:
 (x+y)+z = x+(y+z) for all x,y,z elements of Z

A2
 Addition is commutative:
 x+y = y+x for all x,y elements of Z

A3
 Z has an identity with respect to addition
 (0)

A4
 Every integer x in Z has an inverse with respect to addition...
 (x)

A5
 Multiplication is associative
 (x*y)*z = x*(y*z) for all x,y,z elements of Z

A6
 Multiplication is commutative
 x*y = y*x for all x,y elements of Z

A7
 Z has a identity with respect to multiplication
 (1)

A8
 For all integers x,y,z, x*(y+z) = x*y + x*z
 (distributive laws)









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+

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

Q1
 Exactly one of the following holds:
 a<b
 b<a
 or a=b

Q2
If a>0, then a<0 and if a<0, then a>0

Q3
If a>0 and b>0,then a+b>0 and ab>0

Q4
If a>0 and b<0, then ab<0

Q5
If a<0 and b<0, then ab>0

Q6
If a<b and b<c, then a<c

Q7
If a<b, then a+c < b+c

Q8
If a<b and c>0, then ac<bc

Q9
If a<b and c<0, then ac>bc

A11
 The WellOrdering 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