Class of equivalence

Class of equivalence is mathematical object determined by the specification, in which case the elements of some set \(U\) are considered as "equivalent". The equivalence is denoted with symbol "\(=\)".

It is assumed, that for any objects \(a\), \(b\), \(b\) from \(U\),

if \(a\!=\!b\) , then \(b\!=\!a\);

if \(a\!=\!b\) and \(b\!=\!c\), then \(a\!=\!c\).

Many mathematical objects appear as classes of equivalence of various constructions.

Natural numbers

The natural numbers can be considered as classes of equivalence between sets that, roothily speaking, have equal number of elements.

Two sets \(A\) and \(B\) are considered as equivalent, if there exist function \(F : A \mapsto B\) such that there exist inverse function \(G=F^{-1} : B \mapsto A\) such that \(G(F(a))\!=\!a \forall a\in A\) and \(F(G(b))\!=\!a \forall b\in B\).

However, the elements \(a\) and \(b=F(a)\), equivalent with some class of equivalence, have no need to be equivalent with another class of equivalence. For example, dealing with natural numbers, one may declare that each apple from the desk at the market is "equivalent" to each "quarter coin" in the packet of the customer, although in other senses, an apple has no need to be equivalent (for example, if one bytes an apple).

Basing on the common sense and everyday's observations, in schools, the properties of natural numbers are usually postulated. Then these postulates axioms are used for new constructions and beautiful theorems.

Integer numbers

The integer numbers ca be constructed as class of equivalence of ordered pairs of natural numbers. This can be written in the following form.

\(m\!=\!(a,b)\)

where \(a\) and \(b\) are natural numbers, and \(m\) is integer number.

In order to avoid confusion with other classes of equivalence, the name of class of equivalence can be specified before the parenthesis:

\(m\!=\!\mathrm{integer}(a,b)\)

The special character \(\mathbb Z\) is used to denote the set of integer numbers; \(m \in \mathbb Z\). While working with the same class of equivalence, its name can be omitted.

Using the rules of arithmetics established for the natural numbers, the following class of equivalence is postulated.

Let integer numbers \(m\) and \(n\) are expressed in the following form: \(m\!=\!(a,b)\); \(n\!=\!(u,v)\)

If there exists natural number \(d\) such that \(a\!+\!d = u\) and \(b\!+\!d = v\),

or there exists natural number \(d\) such that \(u\!+\!d = a\) and \(v\!+\!d = b\),

then the integer numbers \(m\) and \(n\) are declared equivalent. The same symbol "-" is used ro denote the equivalence; in this case, one writes \(m\!=\!n\).

Integer number, that can be expressed in form \((a,a)\) for some natural number \(a\) has special name; it is called zero and denoted with character 0.

The operation of summation of the integer numbers \(m\!=\!(a,b)\) and \(n\!=\!(u,v)\) are defined in the following way:

\(m\! + \!n = (a\!+\!u, b\!+\!v)\)

The operation of multiplication of the integer numbers is defined as follows:

\(m\! \times \!n = (a\! \times \!u + u\! \times\!v ~,~ a\! \times \!v + b\! \times\!u )\)

One may check that the tules of arithmetics hold for the operations defined with the formulas above.

For integer numbers, the relation \(>\) and \(<\) can be established in a way, similar to that for natural numbers.

Integer numbers, larger than 0, are considered as equivalent to corresponding natural numbers. In this sense, the set of natural numbers is subset of the set of integer numbers:

\( \mathbb N \subset \mathbb Z\)

This allows to simplify writing of the expressions with integer numbers; the function "minus" is used. Let the integer number \(m=(a,b)\), where \(a\) and \(b\) are natural. Then,

if \(a>b\), then one writes \(m=a\!-\!b\) if \(a=b\), then one writes \(m=0\) if \(a<b\), then one writes \(m=–(b\!-\!a)=0-(b\!-\!a)\)

The last expression can be considered as definition of function \(-\) of single argument. Then the assumption \( \mathbb N \subset \mathbb Z\) allows to use the expression \((a,b)=a\!-\!b\) in all cases, even if \(a\! <\! b\).

Integer number \(m\) with that \(m\!>\!0\) are calld positive, and
Integer number \(m\) with that \(m\!<\!0\) are called negative.

Usually, the symbol \(0\) is omitted in the expressions like \(0\!-\!a\); the oration "-" is interpreted as integer function of an integer argument. Using the special names of the natural numbers, the two-character names can be used:

\(-\!1=0\!-\!1\)

\(-\!2=0\!-\!2\)

\(-\!3=0\!-\!3\)

and so on.

The integer numbers were invented many kiloyears ago; it was great achievement of mathematics, useful for the debit-credit calculation.

The integer numbers are used to construct more complicated objects, and, in particular, the rational numbers.

Rational numbers

Rational numbers are constructed as ordered pairs of integer numbers with certain class of equivalence. The character \(\mathbb {Q}\) is used to denote the set of rational numbers. In general, any rational numbs \(x\) can be written in the following form:

\(x=(m,n)\)

or \(x=\mathbb{Q}(m,n)\)

where \(m\) and \(n\) are integer numbers, and \(n\!\ne\! 0\).

Two rational numbers \(x=(m,n)\) and \(y=(p,q)\) are considered as equivalent, if there exist natural numbers \(c\), \(d\) such that

\( m \times c = p \times d ~\) and

\( n \times c = q \times d\)

The first number (\(m\)) in the pair is called numerator and the second (\(n\)) is called denominator. Rational number that can be written in the form \((0,n)\) has special name, it is called "zero", and the character "0" is used to denote it; it is the same character, used for natural numbers and integer numbers. The set of integer numbers is considered as subset of rational numbers; the following equivalence is assumed:

\( (m,1)=1\)

Summation of rational numbers \(x=(m,n)\) and \(y=(p,q)\) is defined in the following way:

\(x\!+\!y=(m\times q+n\times p, n\times q)\)

Multiplication of rational numbers \(x=(m,n)\) and \(y=(p,q)\) is defined in the following way:

\(x\times y=(m\times p, n\times q)\)

One may check, the rules of arithmetics hold for such definitions. This allows to simplify the representation of pair of number; assuming that \(\mathbb Z \subset \mathbb Q\), the operation slash is defined on the pair of rational numbers,

\((m,n)=m/n\)

The notation with horizontal fraction is also used,

\( \displaystyle (m,n)= \frac{m}{n}\)

Rational numbers are used to built-up the real numbers.

Class of equivalence as general tool

Classes of equivalence allow to create the new objets using the objects with already known properties.

The Real numbers appear as class of equivalence of the Cauchi sequences of rational numbers.

The Complex numbers appear as class of equivalence of the ordered pairs of real numbers and so on.

Classes of equivalence allow to formulate statements, definitions, theorems in general form, applied to pretty different (and sometimes unexpected) cases. For example, one of readers was surprised to see that the Newton method of approximation of the solution of an equation works well not only for real numbers, but also for the complex numbers.

References

http://en.citizendium.org/wiki/Equivalence_relation

http://www.proofwiki.org/wiki/Definition:Equivalence_Class