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In: Statistics and Probability

Define the exponentiation operator on naturals recursively so that x0 = 1 and xS(y) = xy...

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

Expert Solution

ANSWER:

Given that:

he exponentiation operator on naturals recursively so that x0 = 1 and xS(y) = xy · x.

to show that ,

x^y·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,

orchestra answered 1 year ago

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