Alpha-conversion needs to be performed to avoid variable capture. This happens when reducing and the term N has some free variable y that would become bound if N is substituted for x in M because there is a somewhere inside M. To avoid capture of y, in M has to be renamed into some . For example, in problem (a), x would be captured if substituted for y in , so this bound x has to be renamed.

Consider problem (b). I'll omit some parentheses and will write for .

-> (substituting for x)

-> (need to rename y because the argument has y free)

-> (substituting for x)