Counting with Venn Diagrams

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Counting with Venn Diagrams
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Inclusion-Exclusion aka the Sieve Method

• We will denote
• As complement as A.
• A? B as A B
• A ? B as AB

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Inclusion-Exclusion 2 Sets

• A B A B - AB
• A B U - A B A B
• U - A - B AB

A
B
AB
A - B
B - A
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Example 1

• There are 100 students.
• 50 students speak Java.
• 40 students speak C.
• 20 students speak both.
• How many speak neither Java nor C?

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Example 2

• How many arrangements of the letters in
  MISSISSIPPI do not begin with M and do not end
  with I?
• Let U all arrangements of these letters.
• Let A all arrangements that begin with M.
• Let B all arrangements that end with I.
• We want A B U - A - B AB
• P(111, 4, 4, 2) - P(10 4, 4, 2) - P(10 4, 3,
  2, 1) P(9 4, 3, 2)

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Inclusion-Exclusion 3 Sets

• ABC ABC - AB-AC-BC ABC
• ABC U - ABC A B C
• U - A - B - C AB AC BC -
  ABC

A
B
C
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Example 3

• In a state-wide politician survey
• 260 partake of illegal drugs weekly
• 208 pack a gun
• 160 buy sex
• 76 partake and pack
• 48 partake and buy
• 62 pack and buy
• 30 partake, pack, and buy
• 150 do none of the above (yeah, right).

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• How many politicians
• were surveyed?
• are partaking and packing, but not buying?
• are partaking and buying, but not packing?
• are packing and buying, but not partaking?
• Are buying, but neither partaking nor packing?
• Are partaking, but neither packing nor buying?
• Are packing, but neither partaking nor buying?

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Example 4

• There is a small swingers party among prez
  candidates, Gore (G), Bush (B), Trump(T), and
  their wives (i.e., 6 people).
• The wives randomly pick a male partner for the
  remainder of the party (e.g., Tipper swaps Al for
  George, etc.).
• What is the probability that no wife spends the
  party with her hubby?

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• Let G denote all pairings where Gore is with his
  wife.
• Let B denote all pairings where Bush is with his
  wife.
• Let T denote all pairing where Trump is with his
  wife (does he have one currently?).
• G B T U - G - B - T GB GT
  BT - GBT

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• Let N A B C.
• Let I1 A B C.
• Let I2 AB AC BC.
• Let I3 ABC.
• Then, N U - I1 I2 - I3.
• Partition the elements of U according to whether
  they are in K 0, 1, 2, or all 3 sets.

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• For each K value, we see how many times such an
  element is counted by the formula.
• e ? A, B, and/or C are counted 0 times.
• e in none are counted exactly one.
• K U -I1 I2 -I3
• 0 1 0 0 0
• 1 1 -1 0 0
• 2 1 -2 1 0
• 3 1 -3 3 -1

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• ? ? ? ?
• ? ? ? ? ? ? ?
• ? ? ?
• ? ?
• ? ? ? ? ?
• ? ? ? ? ? ? ? ? ?