Question

1. Let g(x) = x2 and h(x, y, z) =x+ y + z, and let f(x, y) be the function defined from g and f by primitive recursion. Compute the values f(1, 0), f(1, 1), f(1, 2) and f(5, 0). f(5, ). f(5, 2)

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Answer 1

As we know recursion refers to function calling itself.

Example x+y

f(x,0)=x

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

f(1,0)=1

f(1,1)=f(1,0)+1=1+1=2

f(1,2)=f(1,1)+1=(f(1,0)+1)+1=1+1+1=3

f(5,0)=5

f(5,1)=f(5,0)+1=5+1=6

f(5,2)=f(5,1)+1=(f(5,0)+1)+1=5+1+1=7

Example g(x)=x^2

Compare x^2 with x^y

x^0=1

x^(y+1)=x^y*x

f(1,0)=1

f(1,1)=x^(0+1)=x^0*x=1*1=1

f(1,2)=x^(1+1)=(x^(0+1))*x=x^0*x*x=1*1*1=1

f(5,0)=1

f(5,1)=x^(0+1)=x^0*x=1*5=5

f(5,2)=x^(1+1)=(x^(0+1))*x=x^0*x*x=1*5*5=25