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 Show that A(m, 2) = 4 whenever m≥1. Hint: Induct on m.

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Here's the Solution to this Question

We shall prove by inducting over m.

When m=1 .

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

It is true for m=0.


Suppose it is true for some m=k\geq1 . We shall prove that it holds for m=k+1 .


A(k+1,2)=A(k,A(k+1,1))\\

Since k\geq1, k+1\geq2 . Hence, A(k+1,1)=2

So,

A(k+1,2)=A(k,2)

We know that it is true for m=k . So, A(k,2)=4 .


Therefore, A(k+1,2)=4.

This shows that it holds for m=k+1 .


Hence, A(m,2)=4 for m \geq1 .


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