Sets N, R, C, Z, and Q

Date: 01/22/2001 at 06:40:57
From: Fenja, Eva & Liliane
Subject: Sets N, R, C, Z, Q

Hello Dr. Math,

Could you help us with these questions?

1) What are the exact and extensive definitions of the sets N, R, C, Z 
   and Q?

2) What proportion do these sets bear to one another?

We thank you in advance. Yours sincerely,

Fenja, Eva and Liliane (from Holland)

Date: 01/22/2001 at 14:09:46
From: Doctor Rob
Subject: Re: Sets N, R, C, Z, Q

Thanks for writing to Ask Dr. Math, Fenja, Eva, and Liliane.

1) The smallest set N that satisfies the following postulates is 
indistinguishable from, and can be taken to be, the natural numbers:

  P1.  1 is in N.
  P2.  If x is in N, then its "successor" x' is in N.
  P3.  There is no x in N such that x' = 1.
  P4.  If y in N isn't 1, then there is a x in N such that x' = y.
  P5.  If x and y are in N and x' = y', then x = y.
  P6.  If S is a subset of N, 1 is in S, and the implication (x in S
       ==> x' in S) holds, then S = N.

This is the recursive definition of addition:

  D+:  Let a and b be in N.
       (1) If b = 1, then define a + b = a' (using P1 and P2).
       (2) If b isn't 1, then let c' = b, with c in N (using P4), and
       define a + b = (a + c)'.

This can be restated as (1) a + 1 = a', and (2) a + b' = (a + b)'.

This is the recursive definition of multiplication:
  D*:  Let a and b be in N.
       (1) If b = 1, then define a * b = a.
       (2) If b isn't 1, then let c' = b, with c in N (using P4), and
       define a * b = (a * c) + a.

This can be restated as (1) a * 1 = a, and (2) a * b' = (a * b) + a.

This is the definition of the order:
  D<:  Let a and b be in N.  a < b if and only if there exist c in N
       such that b = a + c.

It is a long exercise to prove that the natural numbers obey the
commutate, associative, and distributive laws, that 1 is the
multiplicative identity, and that order is respected by addition and
multiplication.

Recall that a relation R on a set S is a subset of the Cartesian
product S x S, that is, R is a subset of the set of all pairs of
elements of S.

Recall that an equivalence relation is a relation which is reflexive,
symmetric, and transitive, that is, satisfies the following three
properties:

   (1) For all a, (a,a) is in R.
   (2) For all a and b, (a,b) in R implies that (b,a) is in R.
   (3) For all a, b, and c, (a,b) and (b,c) both in R implies that
       (a,c) is in R.

For an equivalence relation R we often denote (x,y) in R by x ~ y.

Then the set Z of integers is defined to be the set of all equivalence
classes of pairs of natural numbers (a,b), where the equivalence
relation is

   (a,b) ~ (c,d) if and only if a + d = b + c.

The class containing (1,1) is called "zero" and denoted "0".  The
class containing (a',1) can be identified with the natural number a
in N, and is so denoted.  The class containing (1,a') is called
"negative a" and is denoted by "-a".

One defines addition, multiplication, and the ordering in Z by

   (a,b) + (c,d) = (a+c, b+d),
   (a,b) * (c,d) = (a*c+b*d, a*d+b*c),
   (a,b) < (c,d) if and only if a + d < b + c.

One defines subtraction by a - b = a + (-b).

It is again an exercise to prove that the integers obey the 
commutative, associative, and distributive laws, 1 is the 
multiplicative identity, 0 is the additive identity, -a is the 
additive inverse of a, and that the order is respected by addition and 
multiplication. Furthermore the pair (a,b) is an element of the 
equivalence class a - b.

Then the set Q of rational numbers is defined to be the set of all
equivalence classes of pairs of integers (a,b) with b not zero, where
the equivalence relation is

   (a,b) ~ (c,d) if and only if a*d = b*c.

The class containing (a,1) can be identified with the integer a in Z,
and is so denoted.

One defines addition, multiplication, and the ordering in Q by

   (a,b) + (c,d) = (a*d+b*c, b*d),
   (a,b) * (c,d) = (a*c, b*d),
   (a,b) < (c,d) if and only if 0 < b*d*(b*c-a*d).

It is again an exercise to prove that the rational numbers obey the
commutative, associative, and distributive laws, 1 is the
multiplicative identity, 0 is the additive identity, -a is the
additive inverse of a, if a isn't zero, then 1/a is the multiplicative 
inverse of a, and the ordering is respected by addition and
multiplication.  One defines subtraction by a - b = a + (-b).  One
defines division by a/b = a*(1/b).  Then the pair (a,b) is an element
of the equivalence class a/b.

Then the set R of real numbers is defined to be the set of all
equivalence classes of pairs of subsets of Q (A,B) such that
Q  = A U B, and for every a in A and every b in B, a < b.  Such
pairs are called "Dedekind cuts."  The equaivalence relation is

   (A,B) ~ (C,D) if and only if #(A-C) <= 1 and #(B-D) <= 1.

The rational number a can be identified with the equivalence class
containing the pair (A,B) where A = {x in Q: x < a} and B = Q - A.

It is again an exercise to prove that the real numbers obey the
commutative, associative, and distributive laws, 1 is the
multiplicative identity, 0 is the additive identity, -a is the
additive inverse of a, if a isn't zero, then 1/a is the
multiplicative inverse of a, and the ordering is respected by
addition and multiplication.

Then the set C of complex numbers is defined to be the set of
all pairs or real numbers (a,b), with the following definitions
of addition and multiplication:

   (a,b) + (c,d) = (a+c, b+d),
   (a,b) * (c,d) = (a*c-b*d, a*d+b*c).

The real number a can be identified with the pair (a,0).  The
pair (0,1) satisfies (0,1)^2 = (-1,0) = -1, and so we denote
(0,1) by "i".  We write (a,b) = a + b*i.

It is again an exercise to prove that the complex numbers obey the
commutative, associative, and distributive laws, 1 is the
multiplicative identity, 0 is the additive identity, -a is the
additive inverse of a, and if a isn't zero, then 1/a is the
multiplicative inverse of a.

NOTE: There is an alternative definition of the real numbers as the 
set of equivalence classes of Cauchy sequences of rational numbers.

A sequence of rational numbers is a function f from N to Q.

A Cauchy sequence or rational numbers is a sequence of rational
numbers such that given any e > 0 in Q, there is an x in N such
that for all m > x and n > x in N, |f(m)-f(n)| < e.  In English,
this means that the values of f get arbitrarily close together as
its argument gets large.

Two Cauchy sequences f and g are called equivalent if, for every
e > 0 in Q, there is an x in N such that for all n > x in N,
|f(n)-g(n)| < e.  In English, this means that the values of f
and the values of g get arbitrarily close together as their common
argument gets large.

Any a in Q can be identified with the equivalence class containing
the Cauchy sequence defined by f(n) = a for every n in N.

The Dedekind cut (A,B) can be derived from this by letting A be
the set of rational numbers which are smaller than all but a
finite number of values of the Cauchy sequence f, and B = Q - A.

It is also an exercise to show that each of the containments

   N < Z < Q < R < C

is proper, in that there is always an element in the right-hand set
which is not in the left-hand one.

2) Let #S be the cardinal number of the set S.  Then

   #C = #R,
   #R = (uncountable infinity) * #Q,
   #Q = #Z = #N.

The density of N in Z is 1/2.  The density of Z in Q is 0.  The
density of Q in R is 0.  The density of R in C is 0.

These may not be just what you wanted for the answer to (2), however.

If you need more assistance with this, write again.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/