function application of N x N -> N Let the function cane and its auxiliary helping function down be the smallest functions satisfying the following requirement. For every x∈ℕ, for every y∈ℕ, and for p=(x,y), all of the following statements hold:

down(x+1,y)=(x,y+1)

down(0,y+1)=(y,0)

cane(p)= 0 if down(p) is undefined

cane(p)=1+cane(down(p)) if both down(p) and cane(down(p)) are defined.

(NOTE: p above is a pair.)

For every x ∈ ℕ and y ∈ ℕ such that x+y ≤ 2 (there are six such pairs), calculate the value, if there is one, of down(x,y) and also of cane(x,y). Put your answers in two tables, one for down and one for cane, of this form:



Hint: It might help you for the later parts of this question if you he k these two functions with more values for x and y.

Describe in words how the function cane and its auxiliary helping function down work, and what they accomplish. Explain any wider significance ane these functions might have. My question is how am I supposed to fill in this table;

for example, if I take in x to be 0 and y to be 0 then my down function would be as follows;

down(0+1,0)=(0,0+1)==(1,0)=(0,1) and down(0,0+1)=(0,0)==(0,1)=(0,0) I believe then that when x=0 and y=0 then the position in the table is "undefined" and true for all other coordinates of the table because they form a trail back to (0,0).

Could someone please help me understand this?
A more formal way to define the down function is
$$
\mathit{down}(x,y) =
\begin{cases}
(x-1,y) &amp; \text{if } x &gt; 0, \\
(y-1,0) &amp; \text{if } x = 0 \text{ and } y &gt; 0, \\
\text{undefined} &amp; \text{if } x=y=0.
\end{cases}
$$