| Addition Of Natural Numbers |
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NOTATION AND TERMS The operation of addition, commonly written as the Infix Operator "+", is a Function + : '''N''' × '''N''' → '''N'''. For natural numbers ''a'', ''b'', and ''c'', we write : Here, ''a'' is the ''augend'', ''b'' is the ''addend'', and ''c'' is the ''sum''. DEFINITION We let ''S''(''a'') denote the ''successor of a'' as defined in the Peano Postulates . Addition is defined inductively by fixing the augend. In other words, we let ''a'' be any arbitrary, but fixed natural number, and we then make the following definitions:
By the recursion theorem, this defines a unique function "''a'' +" : N → N. In words, it says that adding zero to ''a'' gives back ''a'', and that applying the successor function to the addend has the effect of applying the successor function to the sum. Since ''a'' was an arbitrary natural number, we can "put together" all these functions into a single binary operation N × N → N. PROPERTIES The following are three immediate and important properties of addition which can be deduced from the definition.
: ( Proof )
: ( Proof )
: ( Proof ) Together, these three properties show that the set of natural numbers N under addition is a commutative Monoid . |
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