Using the definition of the integers as a set of elements which is isomorphic to

Question

Using the definition of the integers as a set of elements which is isomorphic to the equivalence classes of NXN where N is the set of Natural numbers and the equivalence relation is given by (a,b)~(c,d) if a+d=b+c

Prove that the product of a positive integer and a negative integer is negative.

Hints: An integer is associated with the equivalence class (a,b) is positive if a> b and negative if a<b. (a,a) represents the equivalence class for the integer 0. Addition is defined as (a,b) + (c,d)=(a+c, b+d). Multiplication is defined as (a,b)* (c,d)=(ac+bd,ad+bc). The natural number system is closed under multiplication and addition.

Explanation / Answer

Let m=(a,b) with a>b and n=(c,d) with c<d be a positive and a negative integer resply.

Note that a,b,c,d are natural numbers.

Then mn = (a,b).(c,d)

= (ac+bd, ad+bc).

Now a>b , a,b natural numbers => a=b+p for some natural number p.

similarly c<d => d = c+q for some natural number q.

ac+bd = (b+p)c+b(c+q) = 2bc+bq+cp = bc+bc+bq+cp = bc+b(c+q)+cp = bc+bd+cp.

ad+bc = (b+p)d+bc = bc+bd+pd.

Thus since c<d we have cp<dp => bc+bd+cp < bc+bd+pd or in other words ac+bd < ad+bc.

Thus mn is negative.

This completeles the proof.

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