Let g : Nx N → N be the function g(a, b) = 2a + 3b. Use strong induction to prove that if n E N, and %3D n > 10, then n is in the image of g.

Q16

Prove by induction consider an inductive definition of a version of Ackermann’s function. A(m, n)= 2n, if m = 0 0, if m ≥ 1, n = 0 2, if m ≥ 1, n = 1 A(m − 1, A(m, n − 1)), if m ≥ 1, n ≥ 2   1. Find A(1, 1). 2. Find A(1, 3). 3. Show that A(1, n) = 2n whenever n ≥ 1. 4. Find A(3, 4).

Suppose that E is an alphabet, and that f :E* - E* has the property that f(ơ) = o for every o e £ and f(xy) = f(x)f(y) for every x, y e £*. Prove that for every x e E*, f(x) = x.

Let g: N→ N be defined by f(n) = n + 1 for all n E N. Is f surjective?

Consider E={-2+1/n} n=1 to infinity U (3,9) as a subset of R with the usual definition of < (less than) a<b is b-a is postive. List the infimum of E.

Let J5 = {0, 1, 2, 3, 4), and define G: J5 x J5 J5 x J5 as follows. For each (a, b) € J x J₁, G(a, b) = ((3a + 1) mod 5, (2b − 3) mod 5). Find the following. (a) G(2, 2) = (2,0) (b) G(4,1)= (0,4) (c) G(3, 4) (3,3) (d) G(1, 0) (4,2) Need Help? Read It X X X