I know that how the term "property" is defined.
Definition:
An attribute, quality, or characteristic of something.
Like one of the property of addition is "commutativity" which behaves like,
$a+b=b+a$
And similarly associativity,
$a+(b+c)$ = $a+(b+c)$
But I always keep apprehending that How were they found and so properties similar to them?, How were they proved true for all numbers?, Does it need some type of induction proof?
When I asked my cousin about that, he said, "They have no need to be proved since they are properties like you possess and so you can make sense of them". But I replied him, "As far as properties of addition, multiplication are concerned, I can make sense of them but what about these types of properties,
Exponential property: $a^n=\frac{1}{a^{-n}}$.
So he suggested me to ask about it on SE and so I'm doing.
Answer
Your definition of property is more English than mathematical. In mathematics, a property is a formula in logic with a free variable. Any object that can be substituted for the free variable and make the formula 'true' is said to have that property.
For example, the property $E(x)$ meaning $x$ is even could be represented by the formula $\exists n \in \mathbb{N}(2*n = x)$. This formula basically says that there exists a natural number such that two times that number equals $x$. Any even number plugged into this formula for $x$ will make it true, while any other number will not.
Now depending on the context, these properties are simply defined, or require proof.
For example, after constructing the natural numbers and real numbers from within set theory as ajotatxe suggests, in order to demonstrate that the construction is valid, we need to prove that our construction satisfies all the necessary properties.
Or, we could simply state that $R$ is a real closed field. Then we are working under the assumption that every element of $R$ has the properties you describe, and no proof is necessary.
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