Let d n rarr n where dn is the number of the natural number

Question

Evaluated Problems-

1) For each of the following functions, prove or disprove: (a) the function is an injection, (b) the function is a surjection, and (c) the function is a bijection.

a) Let PN* = {S ∈ P(N)| S is finite}. Let f : PN* → (N ∪ {O}) be the function given by f (S) is the number of elements in S, for each S ∈ PN*.

b) der :Z⁴ → Z³ given by der(a, b, c, d)= (3a, 2b, c).

2) Let [0, 1] and [2, 4] be intervals in R. Find a bijection from [0, 1] to [2, 4] and prove that it is a bijection.

3) (Sundstrom) Let d: N → N, where d(n) is the number of the natural number divisors of n. This is the number of the divisors function, which was introduced in the previous problem set. Is d an injection? Is d a surjection? Justify your conclusions.

4) Let f : S → T be a function, and let A be a subset of S. Prove that if f is injective, then f^-1[f[A]] = A.

5) Consider the following proof.

Proposition: Let R* = {x ∈ R| x ≥ 0}. Let f : R → R* be given by f(x) = e^x. Then, f is a surjection.

Proof:

1) Let y ∈ R*.

2) Then, ln(y) ∈ R.

3) Let x = In(y).

4) Then, f(x)= e^ln(y) = y.

5) Thus, for all y ∈ R*, there exists an x ∈ R such that y = f(x).

6) Therefore, f is surjective.

Identify any content-related errors in the proof. If there are none, say so.