Axioms and others

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Algebra

Last updated 1:30 AM on 9/30/22
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26 Terms

1
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a-b=a+(-b)
definition of subtraction
2
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a+(-a)=0
axiom of additive inverse
3
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0+a=a
identity axiom of addition
4
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-(a+b)=-a+(-b)
property of opposite of a sum
5
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-(-a)=a
cancellation property of opposites
6
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a(b+c)=ab+ac
distributive axiom
7
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a+b=b+a
commutative axiom of additon
8
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(a+b)+c=a+(b+c)
associative axiom of addition
9
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a=a
reflexive axiom of equality
10
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if a=b, then b=a
symmetric axiom of equalty
11
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if a=b and b=c, the a=c
transitive axiom of equality
12
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a+b= a real number
closure axiom of addition
13
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a(b)= a real number
closure axiom of multiplication
14
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a(b)=b(a)
commutative axiom of multiplication
15
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(ab)c=a(bc)
associative axiom of multiplication
16
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a*1=a
identity axiom of multiplication
17
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a*0=0
multiplicative property of zero
18
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a(-1)=-a
multiplicative property of -1
19
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(-a)b=-ab
property of opposites in products
20
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a(-b)=-ab
property of opposites in products
21
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(-a)(-b)=ab
property of opposites in products
22
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a*1/a=1
axiom of multiplicative inverses
23
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(ab)1/b=a
axiom of multiplicative inverses
24
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1/-a=-1/a
axiom of multiplicative inverses
25
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1/ab=1/a*1/b
property of the reciprocal of a product
26
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a / b=a*1/b
definition of division