For each of the following functions determine whether they are injective and surjective.

(a) f₁ : N P_fin(P) is defined by f₁(n)=the set of all pP such that p|n (p divides n).

The following notation is used: Given a set A, we denote the set of all finite subsets of A by P_fin(A). Also we denote the set of all positive prime numbers by P.

(b) f₂ : N N Q⁺ is defined by f₂(k, l) = k/l

(c) f₃ : [0,] [0,1] is defined by f₃(x) = cosx.

(d) f4 : (0,1) (0,1) is defined by f₄(x) = x²,x³.

(e) f₅ : P(N) P(N) is defined by f₅(X) = X^c.

(f) f₆ :P(N)P(N)P(N) is defined by f₆(X,Y)=X \ Y.

(g) Given a set X N, define X^e ={nX | n is even}

and X^o ={nX | n is odd} Denote by E the set of all even numbers in N and by O the set of all odd numbers in N. Function f₇ : P(N) P(E) P(O) is defined by

f₇(X) = X^e, X^o.