Define addition of natural numbers as follows: For every x, define x + 1 = x?. F

Question

Define addition of natural numbers as follows:

For every x, define x + 1 = x?.
For every x and y, define x + y? = (x + y)?.

Show that this addition is well-defined. That is, show that for all x and w, the value of x + w is defined.

Explanation / Answer

1. A positive integer is prime if it has exactly two factors. 2. A group is a set together with an associative binary operation such that there is an identity element and every element has an inverse. 3. Let X be a metric space. A subset Y of X is open if for every y in Y there exists d> 0 such that x is in Y for every x with d(x,y) Y is an injection if no element of Y has more than one preimage. That is, f(x)=f(y) ==> x=y. 5. Let A be a set of positive integers and for each n let dn=n-1|A intersect {1,2,...,n}|. That is, dn is the proportion of numbers up to n that belong to A. If dn tends to a limit d as n tends to infinity then A is said to have density d. The lim sup of the dn is called the upper density of A. 6. For html reasons write E for the empty set. Then the number 4 is the set {E,{E},{E,{E}},{E,{E},{E,{E}}}}. 7. Let x and y be mathematical objects. The ordered pair (x,y) is the set {x,{x,y}}. 8. A real number is a partition of the rational numbers into two sets A and B such that every element of A is less than every element of B. 9. A function from A to B is a subset F of the Cartesian product AxB with the property that for every a in A there is exactly one b in B such that (a,b) is in F. 10. The function f(x)=sin(x) is the function f:R--> R defined by f(x)=x-x3/3!+x5/5!-... .

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