Are you sure you want to delete this answer? Yes Sorry, something has gone wrong. We will assume that a compiler is available at run-time. The code will work in 2 modes - tautology or equivalence.

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The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10but also for any other integer base e.

A real number that is not rational is called irrational. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countableand the set of real numbers is uncountablealmost all real numbers are irrational. Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers.

In other words, the field of rational numbers is a prime fieldand a field has characteristic zero if and only if it contains the rational numbers as a subfield.

Finite extensions of Q are called algebraic number fieldsand the algebraic closure of Q is the field of algebraic numbers. The real numbers can be constructed from the rational numbers by completionusing Cauchy sequencesDedekind cutsor infinite decimals.Definition An equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive.

Examples: Let S = ℤ and define R = {(x,y) | x and y have the same parity} i.e., x and y are either both even or both odd. The symbol xR is read the equivalence class of x modulo R or more concisely x from MATH at California Polytechnic State University, San Luis Obispo we might write x Definition Let A be a set and A a family of subsets of A.

The equivalence class of an element a is denoted [a] and is defined as the set [] = {∈ ∣ ∼} of elements that are related to a by ~. An alternative notation [a] R can be used to denote the equivalence class of the element a, specifically with respect to the equivalence relation R.

Feb 01, · About equivalence classes Given an equivalence relation R on a set A, equivalence classes are subsets of A in which all elements are pairwise "equivalent" (where "equivalent" is defined by the relation R).

Equivalence classes are sets of test data that execute a program's instructions in the same manner. For example, the datum 30 belongs to the equivalence class of a program that computes a weekly pay based on 40 or fewer hours per week. an equality-type relation, that is, a binary relation that is reflexive, symmetric, and transitive.

For example, if two geometric figures are congruent or similar or if two sets of objects are isomorphic or equipotent, the figures or sets are equal or identical in some regard.

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