[Game Theory]: Proof

Recall that minimal excludant for a set of non-negative integers S denoted by mex(S) is the smallest non-negative integer which is not in the set S, i.e For example, mex(1,3,5) 0 mex(1,2,4,7} = and so on. Use an inductive argument to show that the Nim-addition (i.e., binary addition without carrying) can be described using the mex in the following way: ab=mex(a0.a1, . . . , a(0-1), 0b, 1b, . . . , (0-1)b} (You need to show that a is excluded from the set S={a0, a1, . . . , a(b-1), 0b, 1b, . . . , (0-1) and that all the non-negative integers smaller than a are included into this set Recall that minimal excludant for a set of non-negative integers S denoted by mex(S) is the smallest non-negative integer which is not in the set S, i.e For example, mex(1,3,5) 0 mex(1,2,4,7} = and so on. Use an inductive argument to show that the Nim-addition (i.e., binary addition without carrying) can be described using the mex in the following way: ab=mex(a0.a1, . . . , a(0-1), 0b, 1b, . . . , (0-1)b} (You need to show that a is excluded from the set S={a0, a1, . . . , a(b-1), 0b, 1b, . . . , (0-1) and that all the non-negative integers smaller than a are included into this set