How various properties of numbers, operations are found? 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. 
 A: Most of calculus books begin introducing one way or another the real numbers. And most of these books incroduce them with a set of axioms, which are the properties that you are talking about.
There are not only operation related properties. There are also order properties and a topological property.
But some books begin with the Peano axioms, which give some very basic properties of the natural numbers. With these Peano's axioms at hand, you can define the addition and the product of natural numbers and prove their properties. That is, what were axioms in the former approach, are theorems now. Then you can build the integer numbers, the rational numbers, and finally the real numbers. But all this construction is rather long, and it is usually omitted, favoring the axiomatic approach.
But things can be done even more from scratch. You can start with a somewhat standarized set of axioms called Zermelo-Fränkel (ZF) and construct the set of natural numbers and prove the Peano's "axioms" (which are theorems now). This is also a long work. Just a tip: the number zero is defined to be the empty set, and the successor of a given natural number $n$ is $n\cup\{n\}$. So, yes, natural numbers are sets.
I'm sorry, but I don't have at hand good bibliography about these constructions. Surely somebody can help with this.
A: 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.
A: A property (also called a "predicate" or "propositional function") is simply something that can be true or false of some objects. In logic, the notation $P(x)$ represents the statement that "object x has property P." We could even have predicates that take multiple arguments, like P(x,y); usually a predicate with more than one argument is called a relation.
Some examples of predicates include:


*

*"$x$ is an even number";

*"$x=0$";

*"$4x-y=73$";

*"the polynomial P has at least 2 real zeroes";

*"$x+y=y+x$".


Of course a statement like P(x) does not have any meaning until we decide on a value for $x$. There are many ways to do this: if we already have an object a, we can set $x=a$ and ask if $P(a)$ is true.
Even if we don't have any specific objects, we can still ask meaningful questions. In this case we can quantify over the range of possible values that our variable can take, whatever those might be. So we can ask whether $P(x)$ holds for all $x$ ("universal quantification"), or we could ask if it is true for some $x$ ("existential quantification").
If we know when a predicate P is true, we can use this fact to deduce other things. For instance, if we have an object a such that P(a) holds, then obviously there is at least one such object - namely a. If we have an object b such that P(b) is false, then we know that not all objects have the property P.
Predicate logic is the logical system that formalizes this reasoning about properties of objects. Statements in predicate logic look like $\forall x \forall y(Sxy \leftrightarrow \forall z(Ezx \to Ezy))$. This is read as, "for all x and y, S(x,y) is true if and only if, for all z, E(z,x) implies E(z,y). 
Another example: $\forall x(Px \to \exists y Sxy)$ means "for all x, if P(x), then there is a y such that S(x,y).
Armed with the rules of predicate logic, and some initial assumptions (axioms) that we decide on beforehand, we can proceed to prove statements about the objects we're studying. 
