Overlapping Sets

Question 1

Set-Matrix

Answer 1

Draw the set-matrix table to tackle questions with 2 sets of dimensions.

THIS IS THE MOST IMPORTANT TECHNIQUE

(Sometimes Venn diagrams suffice for easy questions)

Question 2

Max/Min Questions within Sets

Answer 2

• Test extreme values
• Essentially, create 2 versions of the 2x2 matrix, each testing the min scenario and the max scenario.

Question 3

3 Overlapping Sets:

Basic Formula

Answer 3

Total = n(No Set) + n(Exactly 1 Set) + n(Exactly 2 Sets) + n(Exactly 3 Sets)

When it’s difficult to use these formulas, DRAW the 3-circle Venn diagram and enter values

Question 4

3 Overlapping Sets:

Formula #1

Answer 4

Use a 3-circle Venn diagram

n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C) + NOT(A∩B∩C)

Question 5

3 Overlapping Sets:

Formula #2

Answer 5

Total # of Unique Items = n(A) + n(B) + n(C) - n(Exactly 2) - 2n(Exactly 3) + n(None of the 3)

Notes:

• Use this formula when the number of elements belonging to exactly two of the groups is combined to a single sum (i.e., the problem is giving us the number of elements belonging to “exactly two” of the groups)
• n(A) + n(B) + n(C) repeats the n(Exactly 3) 3 times, so we need to subtract 2 of them to leave 1.
• n(A) + n(B) + n(C) repeats the n(Exactly 2) 2 times, so we need to subtract 1 of them to leave 1.

Question 6

3 Overlapping Sets:

Other Formulas

Answer 6

• Total = n(No Set) + n(At Least One Set)
• n(Exactly 2 Sets) = n(A∩B) + n(A∩C) + n(B∩C) - 3n(A∩B∩C)
• n(At Least 2 Sets) = n(A∩B) + n(A∩C) + n(B∩C) - 2n(A∩B∩C)

Question 7

Be careful with the language

Answer 7

“What is the minimum number of additional small-cap funds from this group that would have to beat the S&P 500 so that over 50% of the small-cap funds beat the S&P 500?”
→ Denominator doesn’t need to increase

“What is the fewest possible number of cookies with both chocolate chips and nuts that would need to be added to that batch so that cookies with both nuts and chocolate chips represented more than 3/5 of all the cookies in the batch?”
→ Denominator has to increase

Question 8

Overlapping Sets: Tricky Data Sufficiency

Answer 8

Out of 200 dog owners, is the number of owners who run their dogs greater than the number of owners who walk their dogs?

1. All of the dog owners who run their dogs also walk their dogs.
2. 75 dog owners walk their dogs.

Answer: A. #1 implies that the number of owners who run their dogs will never be greater than the number of owners who walk their dogs.