Integers: Extending a system using an equivalence relation

The integers are obtained from the natural numbers by insisting that all subtraction problems have answers. This can be done by formally adding additive inverses (the negative numbers) or by thinking of integers as equivalence classes of pairs of natural numbers giving subtraction problems with the same answer.

Definition 55   The number system of integers (written ) is obtained from using an equivalence relation on pairs of natural numbers as follows:

1. the pair $(a,b)$ is equivalent to $(c,d)$ precisely when $a+d=b+c$. We write $[a,b]$ for the equivalence class of $(a,b)$ under this relation.
2. addition is defined by
   
   $$[a,b]+[c,d]=[a+c,b+d]$$
   
   

3. multiplication is defined by
   
   $$[a,b][c,d]=[ac+bd,ad+bc]$$
   
   

4. order is defined by
   
   $$[a,b]<[c,d] \mbox{ if and only if } a+d < b+c$$
   
   

5. the inclusion of into is given by $\iota:\ensuremath{\mathbb{N}}\to \ensuremath{\mathbb{Z}}$ with $\iota(a)=[a,0]$

In order for this definition to make good sense it is important that the proposed equivalence relation is in fact an equivalence and that the addition, multiplication, and order do not depend on the choice of representative of the equivalence clases:

Proposition 104   The relation on pairs of natural numbers given by $(a,b)\approx (c,d)$ if and only if $a+d=b+c$ is an equivalence relation.

(3 Points )

Proposition 105   The addition operation in the definition of does not depend on the choice of representative in the equivalence classes.

(2 Points )

Proposition 106   The multiplication operation in the definition of does not depend on the choice of representative in the equivalence classes.

(2 Points )

Proposition 107   The order relation in the definition of does not depend on the choice of representative in the equivalence classes.

(2 Points )

Proposition 108   The inclusion $\iota:\ensuremath{\mathbb{N}}\to \ensuremath{\mathbb{Z}}$ preserves addition, multiplication, and order.

(3 Points )

Since our object is to extend the natural numbers to a system in which every number has an additive inverse we want to be sure that we do indeed get additive inverses, but also that we have not lost any of the nice properties of the operations that we had before we extended:

Proposition 109   The addition operation on is associative, commutative, and has an identity $[0,0]$.

(3 Points )

Proposition 110   The multiplication operation is associative, commutative, has an identity $[1,0]$, and distributes over addition.

(3 Points )

Proposition 111   The order relation on is transitive: if $[a,b]<[c,d]$ and $[c,d]<[u,v]$ then $[a,b]<[u,v]$.

(2 Points )

Proposition 112   Order on satisfies trichotomy: exactly one of $[a,b]=[c,d], [a,b]<[c,d], [c,d]<[a,b]$ holds.

(2 Points )

Definition 56   An integer $[a,b]$ is called negative if $[a,b]<[0,0]$ and positive if $[a,b]>[0,0]$.

Note that because of trichotomy any integer is either positive, negative, or zero ($[0,0]$).

Proposition 113   If $[a,b]<[c,d]$ then

1. if $[u,v]$ is positive then $[u,v][a,b]<[u,v][c,d]$
2. if $[,u,v]$ is negative then $[u,v][c,d]<[u,v][a,b]$
3. if $[u,v]$ is zero then $[u,v][a,b]=[u,v][c,d]$

(3 Points )

Corollary 114   The product of any two negative numbers is positive.

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Proposition 115   Every integer has an additive inverse.

(2 Points )

These propositions tell us that we have been successful in our attempt to extend the natural numbers to include additive inverses without losing any of the nice algebraic properties of the operations. In addition we have retained most, but not all, of the nice properties of the order. What we lose is the well ordering which made induction such a useful tool for :

Proposition 116   is not well ordered.

(2 Points )

Since this way of thinking about the integers is notationally unfamiliar you should determine how to write $[a,b]$ as a signed natural number (i.e. -3, +2, 0) to make a connection with the usual notation for integers.