recursively give change for the remainder. El Generalissimo does not approve of American capitalist greed. Show that there is a set of coin values for which the greedy algorithm does not always give the smallest possible of coins. (b) Now suppose El Generalissimo decides to impose a currency system where the coin denominations are consecutive powers bº, b¹,b²,..., bk of some integer b≥ 2. Prove that despite El Generalissimo's disapproval, the greedy algorithm described in part (a) does make optimal change in this currency system.

We mentioned that if we want to prove P ≠ NP, we only need to pick up any one NPC problem and prove that polynomial-time algorithm does not exist for the problem. If you want to prove P ≠ NP, select one NPC problem based on your preference and describe your idea of why polynomial-time algorithm does not exist for the problem. It does not have to be a formal proof, a description of your idea would be fine.

How can we solve a and b using direct proof?

Justify (argue, prove, ...) that the following formula is valid Vx A(x) V Vx B(x) → Vx (A(x) V B(x))