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## An equivalence relation on

In the set of ordered pairs of natural numbers we define binary relation:

Proposition 6.2.1   The relation is an equivalence relation in .

Proof.
• (R) For any , we have , thus .
• (S) Let and be two elements of . We have:

• (T) Let , and be three elements of . We have:

By addition we have and by Proposition 1.15 we have , i.e. .

On we define two operations, called addition and multiplication, by the following equations:

For example, and . The multipication can be also denoted without any symbol, like . Propositions 2.2, 2.3, 2.4 and 2.5 are very easy to prove; we leave the task to the reader.

Proposition 6.2.2 (Commutative law)   For any two elements and in , the following hold:
(i)
.
(ii)
.

Proposition 6.2.3 (Associative law)   For any three elements , and in , the following hold:
(i)
.
(ii)
.

Proposition 6.2.4 (Neutral element)   For any the following hold:
(i)
.
(ii)
.

Proposition 6.2.5 (Distributive law)   For any three elements , and in , the following hold:
(i)
.
(ii)
.

In Proposition 2.5 each equation is a consequence of the other beacuse of Proposition 2.2. The addition and the multiplication in are compatible with the equivalence relation:

Proposition 6.2.6   For any four elements , , and in , the following properties hold:

Proof. We prove the first property; we have:

By addition we have , i.e. and this equivalent to Because of the commutativity of the multiplication, we can show only the following property instead of the second equation above: