Constructing numbers from sets

Continued from previous blog: A fast review of set theory; same reference: A Second Course in Analysis by M Ram Murty, Hindustan Book Agency.

Mathematicians and philosophers of the nineteenth century pondered deeply into the nature of a number. The question of “what is a number?” is not a simple one. But since mathematicians decided to give foundations of mathematics using the axiomatic method and sets as the basic building blocks, we are led to define numbers using sets. We follow Richard Dedekind (1831-1916) and Giuseppe Peano (1858-1932) in the following construction. It was as late as 1888 and 1889 when this construction was described in two papers written independently by Dedekind and Peano.

We construct a sequence of sets to represent the natural numbers. As noted earlier, zero is represented by the empty set. We have already described the construction of the natural numbers using the empty set. For each natural number n, the successor of n is denoted by n+1 (and sometimes as n^{'}) and defined as

n \bigcup \{ n \}

Thus, each natural number is a set with n elements, namely,

\{ 0,1,2, \ldots, n-1\}

We designate the set of natural numbers by the symbol \mathcal{N}. (It is a matter of personal convenience whether to include zero as a natural number or not. In this discussion, zero is a natural number. In other settings, it may not be. There is no universal convention regarding this and the student is expected to understand depending on the context. Some authors use the term “whole numbers” to indicate that zero is included in the discussion.)

The arithmetic operations on \mathcal{N} are now defined recursively. Addition is defined as a function from \mathcal{N} \times \mathcal{N} to \mathcal{N}:

+ (m,n) = m+n

where m+n is defined recursively by 0+n=n and m'+n = (m+n)^{'}. A similar definition is given for multiplication x by defining 0 \times n = 0 and

m^{'} \times n = (m \times n) + n

We also define m \times n as simply mn which is the familiar symbology.

An equivalence relation on a set S is a subset R of S \times S satisfying:

  1. (reflexive axiom) (a,a) \in R \forall {x} \in S.
  2. (symmetry axiom) (a,b) \in R \Longleftrightarrow (b,a) \in R.
  3. (transitive axiom) (a,b) \in R and (b,c) \in R implies (a,c) \in R.

The notion of an equivalence relation is an abstaction of our concept of equality, or at least what we implicitly expect of the notion of equality. It is more suggestive to write the equivalence relationn, not as a subset of S \times S as indicated above, but rather more symbolically as \sim that our axioms become:

  1. (reflexive) a \sim a, \forall {a} \in .
  2. (symmetry) a \sim b if and only if b \sim a.
  3. (transitive) a \sim b and b \sim c implies a \sim c.

Equivalence relations play a fundamental role in all of mathematics. They allow us to understand aspects of sets by grouping them using certain properties.

To construct negative integers, we define an equivalence relation on \mathcal{N} \times \mathcal{N}. We write

(m,n)\sim (j,k) \Longleftrightarrow m+k=j+n.

Intuitively, we think of (m,n) as m-n so that it becomes evident that our definition is now in terms of concepts that have been defined earlier. This is very similar to how the ancients worked with negative numberss that appeared in an equation. They usually moved them to the other side so that the equation became an equation of non-negative numbers. However, with our set theoretic definition, we have reached a more fundamental and higher level of abstraction. Thus, with our equivalence relation above on the natural numbers, we define the set of integers as the set of equivalence classes of such ordered pairs. It is now easy to see that the following lemma holds:

Lemma 1.1 If (j,k) is an ordered pair of non-negative integers, then exactly one of the following statements holds:

(a) (j,k) is equivalent to (m,0) for a unique non-negative integer m;

(b) (j,k) is equivalent to (0,m) for a unique non-negative integer m;

(c) (j,k) is equivalent to (0,0).

Sometimes, we denote by |(j,k)| the equivalence class of (j,k). With this lemma in place, we now denote by m the set of pairs of non-negative integers equivalent to (m,0) ; by -m the set of pairs equivalent to (0,m) and by 0 the set of pairs equivalent to (0,0). We denote these equivalence classes by \mathcal{Z}.This gives us set theoretic construction of the set of integers.

We can define the operations of addition and multiplication by setting:

|(j_{1},  k_{1})|+|(j_{2}, k_{2})|=|(j_{1}+j_{2}, k_{1}+k_{2})|

|(j_{1}, k_{1})| \times |(j_{2}, k_{2})| = |(j_{1}j_{2}+k_{1}k_{2}, j_{1}k_{2}+j_{2}k_{1})|

This latter definition is best understood if we recall that the symbol (j,k) represents j-k so that the left hand side of the above equation is

(j_{1}-k_{1})(j_{2}-k_{2}) = j_{1}j_{2}+k_{1}k_{2}-(j_{1}k_{2}+j_{2}k_{1})

One needs to check that these definitions are “well-defined” in the sense that they are independent of the representatives chosen for the equivalence class. This can be done as exercises.

In this way, we have now extended the notion of addition and multiplication from the set of natural numbers to the set of integers. Subtraction of integers can be defined by

|(j_{1}, k_{1})|-|(j_{2}, k_{2})| = |(j_{1}, k_{1})| +(-1)|(j_{2}, k_{2})|

where -1 represents the equivalence class (0,1). All of these definitions correspond to our usual notion of addition, subtraction and multiplication. Their virtue lies in their pure set-theoretic formulation.

We can also order the set of integers in the usual way. Thus,

j_{1}+k_{2} < k_{1}+j_{2} \Longrightarrow |(j_{1}, k_{1})|< |(j_{2}, k_{2})|

and

j_{1}+k_{2} \leq k_{1}+j_{2} \Longleftrightarrow |(j_{1}, k_{1})| \leq |(j_{2}, k_{2})|.

This corresponds to our usual notion of “less than” and “less than or equal to”.

Finally, we can define the absolute value on the set of integers by setting

|k|=k, if 0<k

|k|=0, if k=0

|k|=-k, if k<0

We can now construct the rational numbers \mathcal{Q} from the set of integers. We do this by defining an equivalence relation on the set \mathcal{Z} \times \mathcal{Z}^{+} by stating that two pairs (j_{1}, k_{1}) and (j_{2}, k_{2}) are equivalent if and only if j_{1}k_{2}=j_{2}k_{1}. Intuitively, we think of (j_{1}, k_{1}) as representing the “fraction” \frac{j_{1}}{k_{1}} and examining what we would mean by \frac{j_{1}}{k_{1}} = \frac{j_{2}}{k_{2}} by reducing it to notions already defined. The set of rational numbers \mathcal{Q} is then defined as the set of such equivalence classes.

The expected operations of addition and multiplication are now evident:

|(j_{1}, k_{1})|+|j_{2}, k_{2}| = |(j_{1}k_{2}+j_{2}k_{1}, k_{1}k_{2})|

|(j_{1}, k_{1})||(j_{2}, k_{2})| = |(j_{1}j_{2}, k_{1}k_{2})|

Again, these definitions are easily verified to be well-defined. Finally, we can now define “division”.If |(j_{1}, k_{1})|, |(j_{2}, k_{2})| \in \mathcal{Q} with j_{2} \neq 0, we define:

\frac{|(j_{1}, k_{1})|}{|(j_{2}, k_{2})|} = |(j_{1}k_{2}, j_{2}k_{1})|

These operations satisfy the familiar laws of associativity, commutativity and distributivity. Subtraction of rational numbers then can be written as :

|(j_{1}, k_{1})|-|(j_{2}, k_{2})| = |(j_{1}, k_{1})|+(-1,1)|(j_{2}, k_{2})|

The ordering of rational numbers can also be written as:

|(j_{1}, k_{1})|< |(j_{2}, k_{2})| \Longleftrightarrow j_{1}k_{2}< j_{2}k_{1}

|(j_{1}, k_{1})|   \leq  |(j_{2}, k_{2})| \Longleftrightarrow j_{1}k_{2} \leq j_{2}k_{1}.

These definitions agree with out usual notions of ordering of the rational numbers.

Finally, the definition of absolute value can be extended as:

|[(j,,k)]|= |(j,k)| if [(0,1)] < [(j,k)]

|[(j,k)]| = |(0,1)| if |(j,k)|=|(0,1)|

|[(j,k)]| = -|(j,k)|<  |(0,1)|.

Again, our familiar properties of the absolute value of rational numbers hold. With this foundational construction in place, we can conveniently represent the equivalence class of (j,k) as simply the fraction j/k and continue to work with these numbers as we were hopefully taught from childhood.

In the next sections/blogs we construct the real numbers from this axiomatic framework.

Exercises:

Hint (generic): keep the meaning of the symbols in mind and meaning of equivalence relations and equivalence classes. Also note that our basic object is a class and a set is a member of a class.

  1. let |(j_{1}, k_{1})|, |(j_{2}, k_{2})| be two elements of \mathcal{Z}. Show that the addition:

|(j_{1}, k_{1})|+|(j_{2}, k_{2})| = |(j_{1}+j_{2}, k_{1}+k_{2})| is well-defined. That is, prove that for any (j_{1}^{'}, k_{1}^{'}) \in |(j_{1}, k_{1})| and (j_{2}^{'},k_{2}^{'}) \in |(j_{2}, k_{2})|, we have that (j_{1}^{'}+j_{2}^{'}, k_{1}^{'}+k_{2}^{'}) is equivalent to (j_{1}+j_{2}, k_{1}+k_{2}).

2. For j_{1}, j_{2}, k \in \mathcal{Z}, prove the distributive law: (j_{1}+j_{2}).k = j_{1}k+j_{2}k.

3. Show that the relations < and \leq on \mathcal{Z} have the following properties:

(a) |(0,j)|< |(0,0)| for all j \in \mathcal{Z}^{+}

(b) |(0,j)|< |(k,0)| for all j, k \in \mathcal{Z}^{+}

(c) |(0,j)|< |(0,k)|, j ,k \in \mathcal{Z}^{+} if and only if k<j

(d) |(0,0)| < |(j,0)| for all j \in \mathcal{Z}^{+}

(e) |(j,0)|<|(k,0)|, j, k \in \mathcal{Z}_{\geq 0}if and only if j<k.

(f) |(0,j)| \leq |(0,0)| for all j \in \mathcal{Z}_{\geq 0}.

(g) |(0,j)| \leq |(k,0)| for all j,k \in \mathcal{Z}_{\geq 0}

(h) |(0,j)| \leq |(0,k)| for j,k \in \mathcal{Z}_{\geq 0} if and only if k \leq j

(i) |(0,0)| \leq |(j,0)| for all j \in \mathcal{Z}_{\geq 0}

(j) |(j,0)| \leq |(k,0)| where j, k \in \mathcal{Z}_{\geq 0} if and only if j \leq k.

Cheers,

Nalin Pithwa.

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