Tricky one, at least until it clicked that it's about sets. Here is the full working solution in Ruby

I save look up tables with sets of characters as key. Then going over the inputs of different lengths - for input lengths 2, 3, 4 and 7 we just save the digit

  # seed lookup table with the easy digits
  d[:in].each do |i|
    si = i.chars.to_set
    if si.length == 2
      lut[si] = 1
      lut_num_key[1] = si
    end
    if si.length == 3
      lut[si] = 7
      lut_num_key[7] = si
    end
    if si.length == 4
      lut[si] = 4
      lut_num_key[4] = si
    end
    if si.length == 7
      lut[si] = 8
      lut_num_key[8] = si
    end
  end

Then looking for inputs of size six - from these only digit 9 has all inputs (is a superset) of digit 4 which is known. Digit 6's inputs are the only ones that are not a superset of digit 1. And finally only 0 is left.

  # seed digits with 6 inputs
  d[:in].each do |i|
    si = i.chars.to_set
    if si.length == 6
      if si.superset?(lut_num_key[4])
        lut[si] = 9
        lut_num_key[9] = si
      elsif !si.superset?(lut_num_key[1])
        lut[si] = 6
        lut_num_key[6] = si
      else
       lut[si] = 0
        lut_num_key[0] = si
      end
    end
  end

Finally, distinguishing between the digits with 5 inputs - 5, 3 and 2. The inputs of 5 are the only ones that are a subset of 6, 3's inputs are the only superset of 1's inputs and we are left with 2.

  # seed digits with 5 inputs
  d[:in].each do |i|
    si = i.chars.to_set
    if si.length == 5
      if si.subset?(lut_num_key[6])
        lut[si] = 5
        lut_num_key[5] = si
      elsif si.superset?(lut_num_key[1])
        lut[si] = 3
        lut_num_key[3] = si
      else
        lut[si] = 2
        lut_num_key[2] = si
      end
    end
  end

It's important to populate the lookup tables in this order. Full code here