###### Asked in Math and ArithmeticHuman Anatomy and PhysiologyHeadaches

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# Do left sided identity and right sided inverse suffice to recover both sided axioms in group theory?

## Answer

###### Wiki User

###### August 18, 2007 4:31AM

No. For a counterexample, define a*b=b for all a and b in the group. Then we can pick any e to be the left identity of all the elements. Similarly, any b has the right inverse e because b*e=e. However, (if there is more than one element), this doesn't satisfy the conditions on a group because there is no single (two-sided) identity element. If a*x=a and b*x=b, then x=a and x=b, which obviously can't hold in the general case.

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Postulates
Euclid's 4th postulate states that all right angles are congruent.
This postulate holds in all non-euclidean geometries as well. So
regardless of the geometry (elliptic/Euclidean/hyperbolic) of the
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Postulates
Euclid's 4th postulate states that all right angles are congruent.
This postulate holds in all non-euclidean geometries as well. So
regardless of the geometry (elliptic/Euclidean/hyperbolic) of the
figure, if both are right angles then they are most definitely
congruent.
Postulates are the axioms which define space, these axioms
cannot be proved. Suffice to say it is true because that is part of
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I hope this answers your question.
-Petroz
Postulates
Euclid's 4th postulate states that all right angles are congruent.
This postulate holds in all non-euclidean geometries as well. So
regardless of the geometry (elliptic/Euclidean/hyperbolic) of the
figure, if both are right angles then they are most definitely
congruent.
Postulates are the axioms which define space, these axioms
cannot be proved. Suffice to say it is true because that is part of
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I hope this answers your question.
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Postulates
Euclid's 4th postulate states that all right angles are congruent.
This postulate holds in all non-euclidean geometries as well. So
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figure, if both are right angles then they are most definitely
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Postulates are the axioms which define space, these axioms
cannot be proved. Suffice to say it is true because that is part of
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I hope this answers your question.
-Petroz
Postulates
Euclid's 4th postulate states that all right angles are congruent.
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figure, if both are right angles then they are most definitely
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Postulates are the axioms which define space, these axioms
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http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms
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http://en.wikipedia.org/wiki/Propositional_calculus
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http://en.wikipedia.org/wiki/Peano_arithmetic
I hope I understood your question.
The short answer is "there is no such thing".
I think the questioner may have meant the 5 fundmental laws in
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A1 - for any such real numbers a and b, a+b=b+a, the commutative
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A2 - for any such real numbers a,b and c, a+(b+c) = (a+b)+c, the
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A3 - for any real number a there exists an identity, 0, such
that, a+0 = a, the identity law
A4 - for any real number a there exists a number -a such that
a+(-a)=0, the inverse law
A5 - for any real numbers a and b, there exists a real number c,
such that a+b=c, the closure property.
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examples of logical axioms. I added a related link about ZFC if
you'd like to learn more.
Non-logical axioms, on the other hand, are the axioms that are
specific to a particular branch of mathematics, like arithmetic,
propositional calculus, and group theory. I added links to those as
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