I have been trying to solve this basic FPGA problem with multiplexers, but I cannot garner the meaning of the notation 0/1. Can someone throw some light in this regard? What do notations like 0/1 mean? Is it that the input can be either 0 or 1?
You are correct in your understanding. The [0/1] represents a 1-bit ROM (or Lookup Table). Each of these 1-bit ROMs are programmed by a bit-stream when the device is turned on in order to configure the behaviour of the logic.
In this case all four 1-bit ROMs can be programmed to allow all possible outputs based on the two inputs to be configured.
For example if you set the value in the four ROMs to [0 0 0 1], you would form an AND gate. Set to [0 1 1 1] you form an OR gate. You can make any of the sixteen possible outputs.
In terms of solving the question, I presume you are given more information as to the value in the ROMs.
Yes, the input can be either 0 or 1. This is a lookup table, the fundamental unit of FPGAs; when programming the FPGA you program in what each of these inputs is.
I don't think this question is solvable without more information. You would need to know what at least one of the programmed values is.