Public sector economics: Taxation

The rules of logic are normally used to differentiate between a valid and invalid mathematical arguments and logic is important in the understanding of mathematical reasoning.  Mathematical statements are usually imprecise and ambiguous and are translated into a logical language in order to make them precise. It begins with translating English sentences into expressions that involve logical connectives and proportional variables given that a language can be very ambiguous. Another point is the use of logical reasoning to solve logic puzzles such as representing two people with A and B and then using such when solving a puzzle. An essential step in mathematical argument involves replacing a given statement with another one having the same true value. Therefore, methods that bring about propositions with the same true value in form of a certain compound proposition are applied widely while constructing a mathematical argument.  Compound proposition means that an expression has been formed out of propositional variables by use of logical operates like A and B (Rosen, 2007).

 An always true compound proposition regardless of the truth values of proportional valuables is referred to a tautology, one that is false is referred to as contradiction and one that is neither contradiction nor tautology is called contingency. Logical equivalents refer to compound propositions having same true values in all cases possible.  There two important logical equivalents referred to as De Morgan’s law which indicates how to negate conjunctions and injunctions (Rosen, 2007). Logical equivalents can be used in the construction of further logical equivalents given that a replacement of a proposition in compound proposition can be done using a compound proposition that is equivalent without having to change the truth value of the initial compound proposition. A propositional satisfiability indicates a satisfiable compound proposition where there exists truth values’ assignment to variables that makes it true. Many problems in broad areas can be modeled in terms of propositional satisfiability(Rosen, 2007).

However, propositional logic cannot be used to provide adequate expression of what mathematical statements means in a natural language.  In such a case predicate and quantifiers are used. Predicate logic in expression of various mathematical statements in ways that allow one to reason and see relations between objects (Rosen, 2007).  Statements showing variables such as X>3 are used in mathematical assertions and cannot be either true or false where specifications of the variables is not provided.  In the above statement, X is the subject while 3 is a property that x may have. Quantification refers to using the proportionate function to create a proposition by expressing the extent to which a predicate is factual over an array of elements.  Universal quantification shows that a predicate is true for all the elements being considered. Existential quantification shows that there is a consideration of one or more element for which the predicate is true.  Predicate Calculus refers to an area of logic which deals with quantifiers and predicates. Nested quantifiers involve one quantifier being within a scope of another.   A lot of mathematical statements involve much quantification of various propositional functions that have over one variable.  The order of quantifier is necessary unless the quantifiers are universal or are existential quantifiers (Rosen, 2007). In these quantifiers, mathematical statements whose expression is in English can have logical expressions translations where the statements are first rewritten so that implied quantifiers and a given domain are indicated.  These statements can be negated through a successful application of the rules for statements that involve one quantifier (Rosen, 2007).

Sets are used in grouping together objects. A set refers to objects that are collected in an unordered manner and containing its elements and are best denoted by use of uppercase letters. Most of the problems involve a test of all elements’ combination to see if they can satisfy some given property.  More than one set can be combined in various ways (Rosen, 2007).  The generalization of various sets into arbitrary sets is known as the principle of exclusion-inclusion an essential technique of enumeration.  Showing that one set is a subset of the other is a way of showing that two sets are equal. Cardinality of finite set refers to the number of elements in a given set.   Sequences refer to the elements’ lists that are ordered which are used in discrete mathematics in various ways.  Sequence refers to a discrete structure used in representation of the ordered list (Rosen, 2007).

Likewise, various mathematics concepts are used in the taxation which includes computation and the application of decimals, percents applying data in tables while using reasoning and logic in problem solving. Application of discrete mathematics to taxation involves the modeling of taxation and its redistribution process in a society that is closed. The applied mathematics framework arises from a discrete kinetic approach for particle systems that are active and the framework is expressed by a system of differential equations that are nonlinear.  The rules used in the computation of tax arise from a precedent but not a general principle. The computation involves various elements presented which must be put into sets of data and function before an answer is arrived at. The mathematical approach is also used in the computation of public debts (Debreu, 1985). The functions created in the commutation of tax involves a range of variables which are needed to account for adjustments , deductions , credits , exemptions and the withholding tax.  In addition the relation between taxation and all debt influences involve a straightforward way of determining the interest incomes, interest expense, capital gain or loss from the nominal specifications.   In the computation of the yield from various instruments is also framed by use of the logic rules.   The mathematical approach is also useful in providing proof for the relationship between exogenous variables and endogenous variable while stating the economic laws that are appropriate and sustainable.  The computation of tax has also applied the predictive analytics systems which has enabled tax collection of tax by the tax bureau. The mathematical approach has been used to associates to every taxpayer of a given state and gives a prediction of the expected returns for the entire population of tax payers.  This has been made possible by the application of the various sets of data about each tax payer and forming a logarithm that can be used to show the possible trend in tax payment. Through the construction of logical equivalents, formulas that can be used in the calculation of personal income tax that involves a very large group of people who can be organized into some specified sets (Debreu, 1985). That way, the same taxation is used in for sets that are equal in terms of their income.

References

Rosen, K. H. (2007). Discrete mathematics and its applications: With combinatorics and graph theory. New Delhi: McGraw-Hill Offices.

Debreu, G. (1985). Mathematical economics: Twenty papers of Gerard Debreu. Cambridge [u.a.: Univ. Press.