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A1
- Addition is associative:
- (x+y)+z = x+(y+z) for all x,y,z elements of Z
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A2
- Addition is commutative:
- x+y = y+x for all x,y elements of Z
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A3
- Z has an identity with respect to addition
- (0)
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A4
- Every integer x in Z has an inverse with respect to addition...
- (-x)
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A5
- Multiplication is associative
- (x*y)*z = x*(y*z) for all x,y,z elements of Z
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A6
- Multiplication is commutative
- x*y = y*x for all x,y elements of Z
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A7
- Z has a identity with respect to multiplication
- (1)
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A8
- For all integers x,y,z, x*(y+z) = x*y + x*z
- (distributive laws)
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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+
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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
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Q1
- Exactly one of the following holds:
- a<b
- b<a
- or a=b
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Q2
If a>0, then -a<0 and if a<0, then -a>0
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Q3
If a>0 and b>0,then a+b>0 and ab>0
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Q4
If a>0 and b<0, then ab<0
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Q5
If a<0 and b<0, then ab>0
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Q6
If a<b and b<c, then a<c
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Q7
If a<b, then a+c < b+c
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Q8
If a<b and c>0, then ac<bc
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Q9
If a<b and c<0, then ac>bc
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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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