Consider what would happen if we were to replace the AND gate in the circuit with an OR gate, a NOR gate, and a NAND gate. Identify the single gates that would produce the same outputs as each of these gates when you invert the inputs.

1. define independent-set as the problem that takes a graph g and an integer k and asks whether g contains an independent set of vertices of size k. that is, g contains a set i of vertices of size k such that, for any v and w in i, there is no edge (v,w) in g. show that independent-set is np-complete.