ABCDEFG
1111000
1110100
1000110
0001011
1100011

arbitrary set:

0100001

Now, I'll transpose the matrix, since I think it makes it easier to explain.

  SSSSS
  eeeee
  ttttt
  12345
 +-----
A|11101
B|11001
C|11000
D|10010
E|01100
F|00111
G|00011

So instead of listing the sets horizontally, I list them vertically. Each horizontal row is an element. Internally, each of those horizontal numbers there is stored as a new integer. So instead of storing "1111000" for the set "ABCD", I store "11101" for the set of all sets taht contain "A".