Threesology Research Journal

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Threes Adventurous as of June 20th, 2025

Note: the following information on set theory was generated by an internet query using the terms "set theory involving patterns-of-three." Unfortunately, both AI and Mathematics have foundations based on dualities which is a point that is not taken into consideration and needs to be illustrated as examples of Persistent Dichotomies. Let me provide some examples for those who are not familiar with the idea that major ideas have foundations in the use of dichotomies which is quite problematic when dealing with the circumstance dealing with a very real possibility that human consciousness is developing along a course of tripartition... to the extent the idea of Trichotomization is rarely, if ever taken into consideration as a viable reality of behavior and cognition:

While there is an ongoing search for a Ternary (Trinary) computer language to go along with the sought after quantum model of computing, there is no publicly visible search for a model of Mathematics based on a foundation of three-patterned ideas. Indeed, in the philosophy of Mathematics, there is no active discussions about Mathematic's reliance of dualities and how this, along with the persistence of dualities found in Psychology, are problematic when confronted by the realization that human consciousness... because of an underlying developmental theme of tripartition (List of Threes in Human Anatomy), it is not intellectually unwieldy to consider human consciousness follows a similar stepping stone arrangement. Indeed, if you have a child that experiencing a developmental shift into a perceptual reality of "Trichotomization" yet are subjected to a school system where major subjects have a reliance on using dualities and the social as well as family environments are filled with people who strive to emulate the so-called "real world" (as it is practiced) involving dualities, necessarily so they will have to develop coping mechanisms as an adapted survival strategy... where both good and/or bad efforts can arise... as well as those coping mechanisms which cause internalized health debilitations such as smoking, drinking, drugs, suicide, etc..., as well as externalized efforts such as fighting, quarreling, war, violence, killing, and other destructive-to-others activities.

Multiple answers are generated but they typically involve the same ideas. Let me provide a sampling of what the google AI "person" came up with. (I alternated between the Chrome, Opera, and Avast browsers. Additional browsers might well have produced alternative AI declarations.)

Set theory, while primarily focused on the study of collections of objects called sets, incorporates patterns of three in several ways. These patterns can involve:

1. Ternary Operations and Relations:
   1. Ternary Operations: These are operations that take three elements from a set and combine them to form a single element.
      • Example: In projective geometry, the process of finding a projective harmonic conjugate involves a ternary operation on three points.
   2. Ternary Relations: These are relations that hold between three elements of sets.
      • Example: A ternary relation can describe collinearity of three points in geometry, meaning they lie on the same straight line. Another example involves triples of points in a cyclic order, like hours on a clock face, where a relation R(a, b, c) holds if one passes through b when going from a to c in a clockwise direction.
2. Ordered Triples (3-tuples):
   1. An ordered triple is an ordered collection of three elements, written as (a, b, c).
   2. The order of the elements in a triple matters.
   3. These are used to represent elements in a Cartesian product of three sets, such as A × B × C, which consists of all possible ordered triples where the first element is from A, the second from B, and the third from C.
   4. While the definition of ordered triples can vary, the choice of definition doesn't significantly impact how they function in set theory due to the existence of natural bijections between different definitions.
3. Partitions of Integers (specifically 3):
   1. A partition of an integer 'n' is an expression of 'n' as a sum of positive integers.
   2. The partitions of 3 are 1+1+1, 1+2 (or 2+1), and 3.
   3. The number of partitions of 'k' is denoted by P(k), so P(3) = 3.
4. 3-Partition Problem:
   1. This is a problem in computer science that asks whether a given multiset of integers can be partitioned into triplets that all have the same sum.
   2. It's a strongly NP-complete problem, meaning it's computationally difficult to solve for large sets.
5. Representation of Sets:
   1. Sometimes, to describe a set with an obvious pattern but too many elements to list explicitly, the first few elements (to establish the pattern, often the first three) are listed, followed by an ellipsis (...) and the last element (for finite sets) or no further elements (for infinite sets).
      • Example: The set of lowercase letters of the English alphabet can be written as {a, b, c, ..., z}. The set of natural numbers is represented as {1, 2, 3, ...}.

In summary, while the core of set theory focuses on sets and their elements, patterns of three appear in:

1. Operations and relations: Ternary operations and relations involve relationships or combinations of three elements.
2. Ordered structures: Ordered triples (3-tuples) represent ordered collections of three elements.
3. Combinatorial problems: The partitioning of integers and the 3-partition problem involve grouping elements into sets of specific sizes.
4. Set representation: The convention of listing the first three elements of a set to establish a pattern.

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These are just some examples of how the concept of "three" appears in set theory and related mathematical fields. Set theory provides a fundamental language for expressing and analyzing these patterns and relationships.

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In set theory, patterns of three elements can be seen in various concepts, including:

1. Ordered Triples:
   1. An ordered triple is an ordered collection of three elements, written as (a, b, c).
   2. The order of the elements matters in an ordered triple, unlike a set, where the order of elements doesn't matter.
      • For example, (1, 2, 3) is different from (3, 2, 1).
   3. Ordered triples are used in various mathematical contexts, including defining the Cartesian product of three sets and defining functions.
   4. Ternary Relations:
      1. A ternary relation is a set of ordered triples, where each triple represents a relationship between three elements from possibly different sets.
         • For example, in geometry, the concept of collinearity of three points can be expressed as a ternary relation.
   5. Ternary Operations:
      1. A ternary operation is an operation that takes three elements from a set and combines them to form a single element of that set.
         • Examples include the projective harmonic conjugate in projective geometry and operations in certain algebraic structures like groups.
   6. Partitions of Integers:
      1. A partition of an integer 'n' is a way of writing 'n' as a sum of positive integers, where the order of the terms does not matter.
         • For example, the partitions of 3 are 3, 2+1, and 1+1+1.
   7. Ellipsis for Pattern Indication:
      1. While not a formal set theory concept involving "patterns of three," the ellipsis (...) is used to indicate that a pattern established by the first few elements (often including the first three) of a set continues, especially for large or infinite sets. For instance, {1, 2, 3, ...} represents the set of natural numbers.
   8. Inclusion-Exclusion Principle:
      • The inclusion-exclusion principle provides formulas for calculating the size of the union of sets, extending to patterns of three sets with a specific formula.

In summary, set theory incorporates patterns of three in various ways, from defining ordered triples to exploring ternary relations and operations, and even in more intuitive ways like using ellipses to indicate the continuation of a pattern in set notation.

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Set theory involving patterns of three

Set theory, while fundamentally about collections of objects (sets), incorporates patterns of three in various ways, particularly in:

1. Set Operations:
   1. Union, Intersection, and Difference: The three primary set operations are union (∪), intersection (⩃), and set difference (∖). These operations allow us to combine, find common elements, or identify unique elements between sets, as explained by GeeksforGeeks and ThoughtCo.
   2. Three-Set Scenarios: Problems and proofs often involve applying these operations to three or more sets. For example, the inclusion-exclusion principle for three sets describes the cardinality of their union as the sum of individual cardinalities, minus pairwise intersections, plus the triple intersection.
   3. Venn Diagrams: Visualizing relationships between sets, especially three sets, is commonly done with Venn diagrams. These diagrams illustrate the overlaps and distinctions between the sets, making it easier to understand operations like union, intersection, and difference in the context of three sets.

Ternary Relations:

1. Definition: A ternary relation is a set of ordered triples, analogous to a binary relation being a set of ordered pairs. It involves three elements or entities connected by a relationship. According to Wikipedia, these relations form a subset of the Cartesian product of three sets.
2. Properties: Just as binary relations have properties like reflexivity and symmetry, ternary relations can also exhibit these properties, but defined in a way that accommodates three elements. For example, a ternary equivalence relation needs to satisfy specific symmetry, reflexivity, and transitivity conditions to be considered an equivalence relation.
3. Applications: Ternary relations appear in various mathematical and computational contexts, such as geometry (collinearity of three points), algebra (ternary operations), and computer science (ternary operators).

Ternary Number System:

1. Balanced Ternary: A balanced ternary system uses digits -1, 0, and 1 to represent numbers, potentially simplifying operations and allowing for the representation of both positive and negative numbers without a separate sign.
2. Ternary Representation of Sets: The ternary representation, especially using 0 and 2 digits, is used to describe the Cantor set. OhioLINK ETD notes that each iteration in constructing the Cantor set adds a restriction to the ternary representation of the numbers in the set.
3. In essence, patterns of three emerge in set theory through:
   • The common use of three sets in operations and examples.
   • The definition and properties of ternary relations, which involve ordered triples.
   • Specific mathematical constructions like the Cantor set, which have properties related to ternary representations.

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Set theory involves the study of patterns of three in various ways, often in the context of relationships between sets or within their structures. Here are some key examples:

1. Ternary Relations and Operations:
   1. Ternary Relations: Just as a binary relation is a set of ordered pairs, a ternary relation is a set of ordered triples, a subset of the Cartesian product of three sets (A × B × C). An example is the collinearity of three points in geometry, where the relation holds if three points lie on the same straight line.
   2. Ternary Operations: A ternary operation combines three elements from a set to form a single element of that set. This is an n-ary operation where n=3.
   3. Ternary Equivalence Relations: These are ternary relations analogous to binary equivalence relations, satisfying properties of symmetry, reflexivity, and transitivity defined for triples. Collinearity in Euclidean space is a classic example.
2. Set Operations and Combinatorics:
   1. Inclusion-Exclusion Principle: This rule helps determine the cardinality of the union of three sets (A ? B ? C), accounting for elements belonging to individual sets, pairs of sets, and all three sets to avoid double counting.
   2. Combinatorial Proofs: Sometimes, counting arguments involving combinations or selections of three elements can be used to prove set-theoretic identities.
   3. Unordered Triples: The concept of an unordered triple {a,b,c} is defined as the union of the singletons {a}, {b}, and {c}, illustrating how a set of three elements is constructed from its individual components.
3. Specific Structures:
   1. Ternary Cantor Set: This mathematical construct involves numbers whose base-3 representation excludes the digit 1, demonstrating patterns related to ternary number systems within set theory.
   2. Balanced Ternary Systems: While not strictly set theory, this system uses digits 0, 1, and -1 and allows for representing numbers without a separate minus sign, illustrating alternative numeric structures with a ternary base.
4. Visual Representations:
   1. Three-Circle Venn Diagrams: These diagrams visually represent the relationships between three sets, including their intersections and unions, offering an intuitive way to understand "patterns of three" in terms of how sets overlap and relate to each other.
   2. In summary, patterns of three in set theory can arise from considering:
   3. Relationships involving three elements (ternary relations).
   4. Operations that combine three elements (ternary operations).
   5. Combinatorial arguments that involve counting arrangements of three elements.
   6. Specific sets constructed using ternary properties like the Cantor set.
   7. Visual tools like Venn diagrams that help illustrate the relationships between three sets.

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In set theory, patterns of three often emerge in the context of ternary relationships and set operations involving three sets.

1. Ternary Relationships:
   1. A ternary relation is a set of triples (ordered or unordered) formed from the elements of three sets.
   2. Imagine three sets A, B, and C. A ternary relationship R on these sets would be a subset of the Cartesian product A × B × C, meaning it's a collection of triples (a, b, c) where a is from A, b is from B, and c is from C.
      • Example: In geometry, the collinearity of three points A, B, and C can be expressed as a ternary relation, where the triple (A, B, C) is in the relation if the three points lie on the same straight line.
2. Cardinality: Cardinality in ternary relationships refers to how many instances of one entity can be associated with pairs of the other two entities. Cardinality constraints can take forms like 1:1:1, 1:1:M, 1:M:N, or M:N:P.
3. Set Operations with Three Sets:
   1. Venn Diagrams: Venn diagrams commonly visualize relationships between three sets. They consist of three overlapping circles, where each section represents different combinations of elements belonging to the sets.
   2. Union and Intersection: Operations like union (elements in any of the three sets) and intersection (elements common to all three sets) can be performed using three sets.
   3. Inclusion-Exclusion Principle: The inclusion-exclusion principle finds the cardinality (number of elements) in the union of three sets (A ⋃ B ⋃ C). This involves summing individual set cardinalities, subtracting the pairwise intersections, and adding back the triple intersection to avoid double counting. |A ⋃ B ⋃ C| = |A| + |B| + |C| - |A ⋂ B| - |A ⋂ C| - |B ⋂ C| + |A ⋂ B ⋂ C|
4. Other Relevant Concepts:
   1. Ternary Operations: A ternary operation takes three elements as input and produces one output.
   2. Ternary Equivalence Relations: Similar to binary equivalence relations, ternary equivalence relations satisfy properties like symmetry, reflexivity, and transitivity.
   3. Cantor Set: The Cantor ternary set is constructed by repeatedly removing the middle third of a line segment.
   4. Set theory provides tools and concepts for analyzing and representing relationships and operations that involve patterns of three elements or sets.
   5. These patterns manifest in various areas, from abstract mathematical structures like ternary relations to visual representations like Venn diagrams.

(AI responses may include mistakes.)

Date of Origination: Friday, June 20th, 2025...Approx. 3:00 AM
Initial posting: Friday, June 20th, 2025...1:36 PM