Overlapping Sets

• For problems involving only two categorizations or decisions use the Double-Set Matrix: a table whose rows correspond to the options for one decision, and whose columns correspond to the options for the other decision. The last row and the last column contain totals, so the bottom right corner contains the total number of everything in the problem.
• Make sure the columns and rows correspond to mutually exclusive options for one decision.
• If no amounts are given, pick smart numbers for total, for problems involving percents pick 100, for problems involving fractions, pick a common denominator for the total.
• Read the problem very carefully to determine whether you need to use algebra to represent unknowns.
• You can extended the Double Set Matrix if a decisions requires you to consider more than two options as long as each set of distinct options is complete and has no overlaps.
• Problems hat involve three overlapping sets should be solved using a Venn Diagram. Remember to work from the inside out.
  1) Fill in the innermost circle, items on all three teams.
  2) Fill in circles for items on two teams, remember to subtract the items on all three teams.
  3) Fill in circles for items only on one team, remember to subtract items on two teams and on all three teams.
  To determine the total, just add all numbers together. 
• The union of two sets is the set of all elements that are found in either of the two sets (or in both of them).
• The intersection of two sets is the set of all of the elements that are found in both sets.