In this exercise we set out to prove by induction that #2X=2#X for any finite set X. For more on induction proofs see Appendix A. (a) Write down the elements of 2{R,G,B}. (b) For any natural number n we introduce a statement Sn saying IF#X=n,THEN#2X=2n(Sn) Explain why the formula #2X=2#X holds for any finite set X iff Sn is true for every nN. (c) Prove that if #X>0 then we can find some xX and Y=X\{x} has n1 elements if #X=n. (d) With the assumptions of the previous part, explain why every element of 2X either contains x or does not. (e) Explain why #2X=22#Y because for every element of 2X not containing X we can take the union with {x} to it and get another subset of X. (f) Prove that Sn1 implies Sn, finishing our induction proof. Writing natural numbers in base b. For any natural number b>1 we say nN is written base b if there are c0,c1,c2, such that 0ci