For each of the following functions determine whether they are injective. Given a setA, we denote the set of all finite subsets o fAbyP_fin(A). Also we denote the set of all positive prime numbers byP.

(a)f₁:NP_fin(P) is defined by

f₁(n)=the set of all pPsuch that p | n (p divides n).

(b)f₂:NNQ⁺is defined byf₂(k, l) =k / l

(c)f₃: [0,][0,1] is defined byf₃(x) = cosx.

(d)f₄: (0,1)(0,1) is defined byf₄(x) =x²,x³.

(e)f₅:P(N)P(N) is defined byf5(X) =Xc.

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

(g)Given a setXN, define

X^e={ n X | n is even}

and

X^o ={ n X | n is odd}

Also denote byEthe set of all even numbers inNand byOthe set of all

odd numbers inN. Functionf₇:P(N)P(E)P(O) is defined by

f₇(X) =X^e, X^o.