Give 10examples of axioms
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Here is an example of a set of axioms that form a mathematical object called a "group:"
Suppose there is a set SS and for elements a,b∈Sa,b∈S there is an operation "++" so that a+ba+b is also a member of SS. SS is a group if and only if it obeys three properties:
1. "+" is associative: (a+b)+c=a+(b+c)(a+b)+c=a+(b+c) for any a,b,c∈Sa,b,c∈S.
2. Identity property: there exists an element 0∈S0∈S such that a+0=0+a=aa+0=0+a=a for any a∈Sa∈S.
3. Inverse property: for every a∈Sa∈S there exists b∈S b∈S such that a+b=b+a=0a+b=b+a=0.
Suppose there is a set SS and for elements a,b∈Sa,b∈S there is an operation "++" so that a+ba+b is also a member of SS. SS is a group if and only if it obeys three properties:
1. "+" is associative: (a+b)+c=a+(b+c)(a+b)+c=a+(b+c) for any a,b,c∈Sa,b,c∈S.
2. Identity property: there exists an element 0∈S0∈S such that a+0=0+a=aa+0=0+a=a for any a∈Sa∈S.
3. Inverse property: for every a∈Sa∈S there exists b∈S b∈S such that a+b=b+a=0a+b=b+a=0.
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