PjProblemStrings Sequences Of Triadic Algebra.

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PjProblemsStrings Sequences Of Triadic Algebra

The term *Algebra* was derived from *Al-Jabr wal-Muqobalah*:
*The Compedious Book Of Calculation By Completion And Balancing*

written by the great mathematician *Mohammed ibn Musa al-Khowarizmi* (A.D. 780 - 850).

(a) What is the meaning of *Triadic Algebra*?

(b) Explain the three concepts that constitute *Triadic Algebra*

(c) Describe the PjProblemStrings Sequences that represent

the three concepts of *Triadic Algebra*.

**The strings**:

S_{7}P_{2}A_{2k} (identity. k = 1, 2, 3, ...)

S_{7}P_{3}A_{3k} (force. k =1, 2)

S_{7}P_{4}A_{4k} (motion. k =1, 2, 3,4)

S_{7}P_{5}A_{5k} (change. k =1, 2)

S_{7}P_{6}A_{6k} (change. k =1, 2, 3, 4, 5)

S_{7}P_{7}A_{7k} (change. k =1, 2)

Where k is not a multiplicand.
**The math**:

All Pj Problems are at play:
*identity*, *force*, *motion*, *change*, *grouping/interaction* and *equilibrium*.
**(a)** *Triadic Algebra* implies that three conceptual pillars constitute the foundation of *Algebra*: **variables**, **relations** and **resolutions**.
**(bi) Variables**

All spaces are mathematical spaces. There are seven mathematical spaces ranging from the *empty space* (S_{1}) to the *multi-matter multi-dynamic space* (S_{7}).

Firstly, the keen observer observes the *containership* of a space: is it empty or occupied and what are its dimensions? Secondly, the observer observes the dynamism (*changes*) of the space. The *changes* imply the *variabilities* of the space. It is these *variabilities* that birthed the concept of *variables*. As a result of their observations and experiences with the physical universe, mathematicians have formulated various abstract spaces and abstract occupants of the spaces. For example, *set theory* is an example of such spaces. In the final analysis, the primary problems of interest in a real or abstract space are problems about the nature of its emptiness or its occupancies. The space in focus here is (S_{7}).

Consider an arbitrary space of type S_{7} (multi-matter and multi-dynamic). This type of space has various occupancies and dynamism. For example, a country is of type S_{7}. Mathematicians use letters to arbitrarily represent anything that changes. These letters are called *variables*. Any word can be established as a *variable*. However, the use of letters is customary. So, we can let X be a collection of residents of S_{7}. We can be more specific by letting X be a collection of female residents in S_{7}. The assignment of a population space to a *variable* establishes the *domain* of the *variable*. It is from this *domain* specific values of the *variable* must come from. In *set language*, these values are called *elements* or *members* of the *domain* of the *variable*.
**(bii) Relations**

Natural and man-made *rules* exist in mathematical spaces. The *connections* these *rules* establish between *variables* constitute the *relations* in the space.

Consider figure 127 which illustrates generally known *relations*: relations between parents and children, relations between siblings and relations between spouses or partners. The *connections* are obvious in the diagram. However, the *rules* that established the *connections* are not obvious in the diagram. The *rules* that established the *connections* illustrated in figure 127 are natural biological processes that humans have defined and assigned terminologies. For example, the terms *mother*, *father*, *son*, *daughter* are man-made formulations from the language used to define the biologic process of conception and birth. It is often the case that natural processes in a mathematical space are defined by the cognitive beings in the space (humans in the case of earth). The concept of definitions of observable processes and their verifications birthed *science*.

The *connections* of primary interest in *Triadic Algebra* are those established by *same origin rules* and *causation rules*. For example, the *connection* of the siblings in figure 127 were established by both the *same origin rule* (all siblings *originate* from the same parents) and the *causation rule* (the respective eggs of their mother and sperms of their father *caused* their respective *beings*). However, it is not always the case that both *rules* are simultaneously applicable in a given *connection*. For example, the *connection* between the mother and father in figure 127 is not established by the *same origin rule*. Rather, it is established by the *causation rule* (the formal or informal agreement between them caused their *relation*).
*Group relations*, *equalities and inequalities relations* constitute the majority of the *relations* between variables in a mathematical space. These *relations* are primarily established by *same origin rules* and *causation rules*. For example, the *domain space* of variables is a consequence of the *same origin rule* and *causation rules* establish functions (important subsets of *relations* between variables in a mathematical space).

functions are very important *relations* in mathematical spaces. The key components are:

- The *domain* D, of the function where the *values of the independent variables* reside.

- The *range* R, of the function where the *values of the dependent variables* reside.

- The *rule* *f* which determines a *unique* element *f(x)* in R, for every *x* in D.
**(biii) Resolutions**
*Resolutions* in the context of *Triadic Algebra* simply mean *finding answers* to the *relations* in mathematical spaces. The *answers* range from general properties of *relations* to specific values of the variables of *relations*. For example, the properties of the *number line*, *trigonometry functions*, *logarithmic functions*, *exponential functions*, *polynomials*, *normal curves*, etc were determined through *resolutions*. It is also through *resolutions* that the specific values of dependent variables in equations and inequalities are determined for given independent variables.

The components of *resolutions* are *mathematical methods* that are *algorithmic* sequences of calculations that lead to the answers sought while obeying mathematical axioms, definitions and rules. Mathematics is replete with calculations. However these calculations do not constitute the *essence* of mathematics. They *resolve* the *essence* of mathematics inorder to satisfy the human need to know by finding answers to the *variabilities* and *relations* in mathematical spaces.
**(c)**

Generally, all PjProblems are at play in a dynamic space. However, the PjProblems in focus differ from scenarios to scenarios. They are the *PjProblems of Interest*. The three PjProblemStrings that represent the three components of *Triadic Algebra* are:
**Variables - S _{7}P_{2}A_{2k}**: here the focus is on the

So,

The PjProblemString Sequence of Interest is the PjProblemString Sequence

S

In other words, when answers are sought in a mathematical space:

First

Then determine the

Then

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

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