Are x⋅0=0, x⋅1=x, and −(−x)=x axioms?

Question

Are x⋅0=0, x⋅1=x, and −(−x)=x axioms?

Lindsey Gamble · Accepted Answer

Step 1  The question is more profound than is initially seems, and is really about algebraic structures. The first question you have to ask yourself is where youre

usumbiix · Accepted Answer

I will assume this is in the context of rings (e.g., real numbers, integers, etc). In this case, the axiom defining 0 is that

x + 0 = x

for all x.

x \times 0 = 0

is a result of this since we have

x \times 0 = x \times (0 + 0) = x \times 0 + x \times 0

which implies

x \times 0 = 0

(canceling one of the

x \times 0^{'} s

).
I am guessing that for the second one you mean

x \times 1 = x

.
This is a definition (axiom).
The third one is a consequence of the definition of -x being the element such that

x + (- x) = 0

.
For then we have

(- x) + x

is also zero so that x is the negative of -x.

alenahelenash · Accepted Answer

Step 1 Let A be a ring (which I will assume commutative). Only the second sentence is an axiom: multiplication has a (provably unique) neutral element, id est, a∈A such that ax=xa=x for every x∈A; its

Are x\cdot0=0,\ x\cdot1=x, and -(-x)=x axioms?

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