Abstract Algebra Theory and Applications - Computer Science ...

12.3 BURNSIDE’S COUNTING THEOREM 215Table 12.1. Switching functions in two variablesInputsOutputsf 0 f 1 f 2 f 3 f 4 f 5 f 6 f 70 0 0 0 0 0 0 0 0 00 1 0 0 0 0 1 1 1 11 0 0 0 1 1 0 0 1 11 1 0 1 0 1 0 1 0 1InputsOutputsf 8 f 9 f 10 f 11 f 12 f 13 f 14 f 150 0 1 1 1 1 1 1 1 10 1 0 0 0 0 1 1 1 11 0 0 0 1 1 0 0 1 11 1 0 1 0 1 0 1 0 1Consider a switching function with three possible inputs, a, b, and c.As we have mentioned, two switching functions f and g are equivalent if apermutation of the input variables of f gives g. It is important to noticethat a permutation of the switching functions is not simply a permutation ofthe input values {a, b, c}. A switching function is a set of output values forthe inputs a, b, and c, so when we consider equivalent switching functions,we are permuting 2 3 possible outputs, not just three input values. Forexample, each binary triple (a, b, c) has a specific output associated with it.The permutation (acb) changes outputs as follows:(0, 0, 0) ↦→ (0, 0, 0)(0, 0, 1) ↦→ (0, 1, 0)(0, 1, 0) ↦→ (1, 0, 0).(1, 1, 0) ↦→ (1, 0, 1)(1, 1, 1) ↦→ (1, 1, 1).Let X be the set of output values for a switching function in n variables.Then |X| = 2 n . We can enumerate these values as follows:(0, . . . , 0, 1) ↦→ 0(0, . . . , 1, 0) ↦→ 1(0, . . . , 1, 1) ↦→ 2(1, . . . , 1, 1) ↦→ 2 n − 1..