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Three-Set Venn Diagrams

Fill every region of a three-circle Venn diagram from survey totals — including “only”, “both but not the third”, and “none”.

Seven Inside Regions Plus Outside

With three sets A, B and C inside a universal set U, the diagram has seven numbered regions inside the circles, plus the outside “none of these”: 1 Only A 2 Only B 3 Only C 4 A \cap B but not C 5 A \cap C but not B 6 B \cap C but not A 7 A \cap B \cap C (all three) Survey questions usually give totals like “ 45 in Music” and “ 15 in both Music and Drama”. Those totals include the triple overlap — so you work from the centre outwards.

Example 1 — Example — fill a three-activity survey

Among 100 students: - Music M = 40, Drama D = 35, Art A = 30 - M \cap D = 12, M \cap A = 10, D \cap A = 9 - M \cap D \cap A = 5 Step 1 — centre: all three = 5 Step 2 — pairwise crescents: - M \cap D not A = 12 - 5 = 7 - M \cap A not D = 10 - 5 = 5 - D \cap A not M = 9 - 5 = 4 Step 3 — only one activity: - Only M = 40 - 7 - 5 - 5 = 23 - Only D = 35 - 7 - 4 - 5 = 19 - Only A = 30 - 5 - 4 - 5 = 16 Step 4 — at least one activity: 23 + 19 + 16 + 7 + 5 + 4 + 5 = 79 None: 100 - 79 = 21

Once the regions are filled, exams ask: How many in none of the activities? \to total - (sum of all seven inside regions) How many in Music only ? \to the only-M region How many in Music and Drama but not Art ? \to the M \cap D crescent How many in Art and Drama but not Music ? \to the A \cap D crescent

Practice

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M \cap D \cap A = 8 and “both Music and Drama” = 15 . How many are in Music and Drama but not Art?

Your answer

You already know every inside region. How do you find “none of the three”?