Solution to Define a function A: N x N -> N as follows: A(m, n) ={2n, if … - Sikademy
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Archangel Macsika

Define a function A: N x N -> N as follows: A(m, n) ={2n, if m= 0; 0, if m≥1 and n= 0; 2, if m≥1 and n= 1; A(m-1, A(m, n-1)), if m≥1 and n≥2 (a) Calculate the following: (i)A(1,0) (ii)A(0,1) (iii)A(1,1) (iv)A(2,2).

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Consider the function A:\mathbb N\times\mathbb N\to\mathbb N,

A(m,n)=\begin{cases} 2n,\ \ \ m=0\\ 0,\ \ \ \ \ m\ge 1\text{ and } n=0\\ 2, \ \ \ \ \ m\ge 1\text{ and } n =1\\ A(m-1,A(m,n-1)),\ \ \ \ m\ge 1\text{ and } n\ge 2 \end{cases}

(i) Since m=1 and n=0 we use the formula A(m,n)=0, and have that A(1,0)=0

(ii) Since m=0 we use the formula A(m,n)=2n, and have that A(0,1)=2

(iii) Since m=1 and n=1 we use the formula A(m,n)=2, and have that A(1,1)=2

(iv) We use the formula A(m,n)=2 for m=2 and n=1 to conclude that A(2,1)=2.

In the following we use the formula A(m,n)=A(m-1,A(m,n-1)) to calculate A(2,2) and A(1,2), and the formula A(m,n)=2n to calculate A(0,2):

A(2,2)=A(1, A(2,1))=A(1,2)=A(0, A(1,1))=A(0,2)=4

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