Question

How do I solve this? Please include clear instructions as I don't have much experience with functions and relations :(

Answer 1

9

Explanation:

From 3),

`A(2,3)=A(2-1,A(2,3-1))=A(1,A(2,2))`

Consider A(2,2). From 3),

`A(2,2)=A(2-1,A(2,2-1))=A(1,A(2,1))`

Consider A(2,1). From 3),

`A(2,1)=A(2-1,A(2,1-1))=A(1,A(2,0))`

Form 2),

`A(2,0)=A(2-1,1)=A(1,1)=\{{\text{Use 3)}}\}=A(1-1,A(1,1-1))=A(0,A(1,0))=\{\text{Use 2)}\}=A(0,A(1-1,1))=A(0,A(0,1))=\{\text{Use 1)}\}=A(0,1+1)=A(0,2)=\{\text{Use 1)}\}=2+1=3`

Then

`A(2,1)=A(1,3)=\{{\text{Use 3)}}\}=A(1-1,A(1,3-1))=A(0,A(1,2))=\{{\text{Use 3)}}\}=A(0,A(1-1,A(1,2-1)))=A(0,A(0,A(1,1)))=\{\text{A(1,1)}=3 \}=A(0,A(0,3))=\{{\text{Use 1)}}\}=A(0,3+1)=A(0,4)=\{{\text{Use 1)}}\}=4+1=5`

Then

`A(2,2)=A(1,5)=\{{\text{Use 3)}}\}=A(1-1,A(1,5-1))=A(0,A(1,4))=\{{\text{Use 1)}}\}=A(1,4)+1=\{{\text{Use 3)}}\}=A(1-1,A(1,4-1))+1=A(0,A(1,3))+1=\{{\text{Use 1)}}\}=A(1,3)+1+1=A(1,3)+2=\{{\text{Use 3)}}\}=A(1-1,A(1,3-1))+2=A(0,A(1,2))+2=\{{\text{Use 1)}}\}=A(1,2)+1+2=A(1,2)+3=\{{\text{Use 3)}}\}=A(1-1,A(1,2-1))+3=A(0,A(1,1))+3=\{{\text{Use 3)}}\}=A(1,1)+1+3=\{A(1,1)=3\}=3+4=7`

Hence,

`A(2,3)=A(1,7)=\{{\text{Use 3)}}\}=A(1-1,A(1,7-1))=A(0,A(1,6))=\{{\text{Use 1)}}\}=A(1,6)+1=...=1+1+7=9`