PjProblemStrings Sequences Of Triadic Algebra.
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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:
S₇P₂A_2k (identity. k = 1, 2, 3, ...)
S₇P₃A_3k (force. k =1, 2)
S₇P₄A_4k (motion. k =1, 2, 3,4)
S₇P₅A_5k (change. k =1, 2)
S₇P₆A_6k (change. k =1, 2, 3, 4, 5)
S₇P₇A_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 (S₁) to the multi-matter multi-dynamic space (S₇).

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 (S₇).

Consider an arbitrary space of type S₇ (multi-matter and multi-dynamic). This type of space has various occupancies and dynamism. For example, a country is of type S₇. 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 S₇. We can be more specific by letting X be a collection of female residents in S₇. 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 - S₇P₂A_2k: here the focus is on the identities of the variablilities in the mathematical space.
Relations - S₇P₆A_6k: here the focus is on the relations of the variables in the mathematical space.
Resolutions - S₇P₇A_7k: here the focus is on finding answers to the relations of the variables in the mathematical space. This was referred to by the great mathematician Mohammed ibn Musa al-Khowarizmi as calculation by completion and balancing. In essence the determination of relational equilibrium.

So, S₇P₂A_2kS₇P₆A_6kS₇P₇A_7k is the PjProblemStrings Sequence of Interest that represent the three concepts of Triadic Algebra. Implicit in the
The PjProblemString Sequence of Interest is the PjProblemString Sequence
S₇P₃A_3kS₇P₄A_4kS₇P₃A_3k.

In other words, when answers are sought in a mathematical space:
First identify the variables of interest.
Then determine the relations (grouping/interactions) of the variables.
Then resolve by finding the answers inherent in the relations through appropriate mathematical methods (i.e. bring to equilibrium).

Blessed are they that have not seen, and yet have believed. John 20:29

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