In: Statistics and Probability
Define the exponentiation operator on naturals recursively so that x0 = 1 and xS(y) = xy · x. Prove by induction, using this definition, that for any naturals x, y, and z, xy+z = xy · xz and xy·z = (xy
ANSWER:
Given that:
he exponentiation operator on naturals recursively so that x0 = 1 and xS(y) = xy · x.
to show that ,
xy·z = (xy)2 we will prove this by induction
Let z =1
result holds for Z =1
Now assume that result holds for z = k
Now to show that result holds for Z = k + 1 cosider.
result holds for Z = k + 1
By induction,
to show that
we will prove this by induction on z .
Let Z = 1
Result holds for z = 1.
Now , assume that result holds for Z = K
to show that result holds for Z = + 1.
consider,
Result holds for z = K + 1.
By induction,