### The True Definition of the Prime Number P # The True Definition of the Prime Number P

Proptety is one of the lesser known but most important components to defining a set. For some sets, each member has a pre-determined meaning and relation to others, while for other sets each member has an independent meaning. In the former cases, there is no such thing as pre-determined meaning, but rather a word or phrase which is dependent on the context of usage to determine its meaning. A set like this could be called a ‘comparable’ set, where there are no pre-determined commonalities among members.

In English, the definition of a word as ‘being equal to another’ is ‘likeness under a definition.’ By using a similar definition, we can infer that there are no differences in X and Y if we use ‘X’ and ‘Y’ to mean ‘the same thing.’ The ‘equivalence’ of terms thus comes into play in mathematics. The set that is Propety is different from the other types of sets considered in this text because it has a unique definition. We now have four different forms of equality:

For any pair P and Q, if P is a power and Q is a non-power, then the members of the P must also be powers. This shows that there are no differences in the members of the P and Q set, i.e., there is no difference in their cardinality, essential property, or symmetric properties. This shows that the definition of ‘P’ is still true and this proves that there are still true definitions for every type of set.

Let us now define the second equality: P is equal to Q if and only if both sides of the equation are real and the values of P and Q are real. This shows that the definition of ‘P’ is dependent on the dimension of the set in question and not on the dimension of the quotient of P and Q. By adopting this approach, it can be shown that the ‘p’ variable can be compared with any other variables without using any concept of identity, i.e., a scalar. This is indeed an important point because it shows that there are no essential axioms for the prime group, hence no necessary connections between any of its components. It can also be derived by applying the axiom of substitution. Thus, we get: P equals Q if and only if the values of P and Q are real.

The third fundamental rule of Propety is the correspondence property. According to this one, all numbers can be used as values of P, provided that they are positive. It can also be derived by applying the axiom of substitution: if a number is substituted for another number, the other one is not changed. Thus, it shows that the definition of the prime number (the original number) is also true for all numbers and all other numbers can also be used as values of P, provided that they are positive. This also results in axiomatrization, which shows that there are no essential connections among any of the elements of a set, thereby eliminating the problem of incompleteness.

The last fundamental rule of Propety considers the domain of numbers as equal to the set-block identity. To see the consistency of this rule, consider the idea that all numbers are real. Then, let us assume that a set has no elements. Let us then consider the equality of such a set. We will still be left with the proposition that P equals Q if and only if the values of P and Q are real.