Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsThe theory of inference for statement calculus in reference of composition mappings and relation matrix
Volume 3(2) · Pages 70–74
Research areasOptics and Image AnalysisGeographic Information Systems StudiesAdvanced Computational Techniques and Applications
Abstract
Author-writtenBIn this article we prove that set of Complex numbers C= a+ib ; where a,b є R and i= imaginary number also know as by iota (i= √-1), is the set of largest number, while set of Natural numbers N={1,2,3….n……} is the set of least numbers by compositions of mappings or product of mappings. We verify the inclusion relation N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C or C⊃R ⊃ Q ⊃ I ⊃ W ⊃ N by relation matrix and graph of relation.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Discrete functions, which transform one finite set into another, are a specific type of relation and are essential for the implementation of computer programs. View evidence ↗
- In computing, the output of a system is viewed as a function of its input. View evidence ↗
The question
- The authors aim to prove that the set of complex numbers (C) is the largest set of numbers and the set of natural numbers (N) is the least set of numbers using compositions of mappings. View evidence ↗
What they did
- The study utilizes the standard sets of numbers: Natural (N), Whole (W), Integers (I), Rational (Q), Irrational (Ir), Real (R), and Complex (C). View evidence ↗
- The authors define a series of identity mappings (f1 through f5) that map each set into the next larger set in the sequence N → W → I → Q → R → C. View evidence ↗
- The research method involves calculating the composition of these five mappings to establish a relationship between the smallest and largest sets. View evidence ↗
- The authors use relation matrices and graphs of relations to further prove the hierarchy of the number sets. View evidence ↗
What they found
- The composition of mappings demonstrates that every natural number can be mapped to a unique image in the set of complex numbers. View evidence ↗
- Using the relation matrix, the authors find that the relation is transitive, confirming that natural numbers are included in the set of integers. View evidence ↗
- The transitive property of the relation matrix further shows that natural numbers are included in the set of rational numbers. View evidence ↗
- By combining the results of the mappings and relation matrices, the authors establish the inclusion chain N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C. View evidence ↗
What it means
- The authors conclude that the set of complex numbers is the largest and the set of natural numbers is the least among the studied number sets. View evidence ↗
Visual summary
1 visualsIllustrations explain the paper; they do not show study data.
Number Set Hierarchy
AI-generated conceptual illustration (nested geometric circles representing mathematical number sets). It explains the idea and does not show study data.
View source evidence ↗Introduction
We know function is a particular class of relations we are primarily concern with discrete functions which transform a finite set into another finite set. There is several such transformations involved in the computer implementation of any program. Computer output can be considered as a function of the input. Let X and Y be any two sets. A relation f from X to Y is called a function if for every x є X there is a unique y є Y such that (x ,y) є f. Note that the definition of function requires that a relation must satisfy two additional condition in order to qualify as a function. The first condition is that every x є X must be related to some to some y є Y, that is domain of f must be X and not merely a subset of X. The second requirement of uniqueness can be expressed as (x, y) є f (x, z ) є f ⇒ y= z. Terms such as "transformation" "map" or ("mappings") "correspondence" and "operation" are used as synonyms for "function". The notations f : X→ Y or X → f Y are used to express f as a function from X to Y. Pictorially a function is generally shown as For any function f: X→ Y, if (x, y), then x is called an argument and corresponding y is called the image of x under f. Instead of writing (x, y) є f, it is customary to write y= f(x) and call y the value of the function f at x. Other ways of expressing y=f(x) are f: X→ Y or X → f Y and of curse (x, y) є f. We sometimes denote the range of f, viz. Rf by f(x). The range of f is defined as {y: ∃ x є X y = f(x)} It was mentioned that the domain of f is X, that is Df = X. The range of f is denoted by Rf and Rf ⊆Y. The set Y is called the co domain off .Since a function is a relation, we can use a relation matrix or graph to represent it in some cases. Note that from the definition of a function it follows that every row of its relation matrix must have only one entry which is 1 while all other entries in the row are 0's. Therefore one can replace the relation matrix by a single column i.e. a vector consisting of entries which are images of the arguments. Thus the column consists of entries which show a correspondence between the argument and the function under the argument. A function can often be represented in graphical form.
Definition 1.6:
A mapping which is one-one as well as onto is called one-one onto mapping it is also called bijective mapping.
Definition 1.7:
A mapping which is many one as well as into is called many one into mapping.
Definition 1.8:
A mapping which is many one as well as onto mapping is called many-one onto mapping. Definition 1.9: If X be a set and f : X→ X is defined by f(x)=x, that is f assign to each element to each element in X the element itself, then f is called the identity mapping on X.
It is denoted by IA. Thus IA: X→X is given by IA(x)= x ∀ x є X. Definition 2.0: Let X, Y, Z be three non-empty sets. Consider tow mappings f: X→ Y and g: Y→ Z, and consider two mappings f : X→ Y defined by y=f(x) where x є X, y є Y and g : Y→ Z defined by z=g(y) where y є Y and z є Z. The composition of mappings f and g (in this order) is the composite mapping denoted by gof and is a function f : X→ Z denoted by gof : X→ Z where (gof)(x)= g{f(x)}, ∀ x є X. Clearly we observed that N,W,I,Q,R,C all are six non-empty set and f1 is a mapping N into W, f2 is a mapping from W into I, f3 is mapping from I into Q,F4 is mapping from Q into R and f5 is a mapping from R into C. Now we shall obtained composition of all these mappings f1, f2, f3, f4, f5.
Results and Discussions (i)
Now we shall find out compositions of all these mappings f1 ,f2, f3 ,f4 f5. We have shown that every natural numbers is mapped on set of complex numbers according to our rule defined by f(x)=x.
Hence for every natural number N there exist unique image in
Acknowledgement
AcknowledgementThe authors are thankful to Dr. K.N. Rajeshwari Professor &HOD School of Mathematics DAVV, Indore for her useful suggestions also thanks Professor Ashutosh Gadgil faculty SVIM and Ms Yashmita Tiwari for drawing the figure.
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