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Logic

Venn Diagrams

Lesson

A Venn diagram shows sets as overlapping circles. Each region represents a distinct combination.

Two-set diagram regions

  • Only A: |A| − |A ∩ B|
  • Only B: |B| − |A ∩ B|
  • Both: |A ∩ B|
  • Neither: |U| − |A ∪ B|

Inclusion-exclusion (3 sets)

∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|

Standard 2-set example

50 students. 30 like math, 25 like science, 15 like both.

  • Like math only: 30 − 15 = 15
  • Like science only: 25 − 15 = 10
  • Like neither: 50 − (30 + 25 − 15) = 10

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

50 students; 30 math, 25 sci, 15 both. |math|?\text{50 students; 30 math, 25 sci, 15 both. |math|?}

Problem 2

Same. |science|?\text{Same. |science|?}

Problem 3

Same. Both?\text{Same. Both?}

Problem 4

Same. Math OR science?\text{Same. Math OR science?}

Practice

Standard problems matching the lesson.

Problem 5

Same. Math only?\text{Same. Math only?}

Problem 6

Same. Science only?\text{Same. Science only?}

Problem 7

Same. Neither?\text{Same. Neither?}

Problem 8

Same. Not math?\text{Same. Not math?}

Problem 9

Same. Like at least one?\text{Same. Like at least one?}

Problem 10

60 surveyed: 35 N, 20 T, 10 both. Only newspaper?\text{60 surveyed: 35 N, 20 T, 10 both. Only newspaper?}

Problem 11

Same. Only TV?\text{Same. Only TV?}

Problem 12

Same. Neither?\text{Same. Neither?}

Problem 13

Same. At least one?\text{Same. At least one?}

Problem 14

Same. Not TV?\text{Same. Not TV?}

Problem 15

3-set: U=100. |A|=50, |B|=40, |C|=30, |A∩B|=15, |A∩C|=10, |B∩C|=8, |A∩B∩C|=5. |A∪B∪C|?\text{3-set: U=100. |A|=50, |B|=40, |C|=30, |A∩B|=15, |A∩C|=10, |B∩C|=8, |A∩B∩C|=5. |A∪B∪C|?}

Problem 16

Same. Number in none of A, B, C?\text{Same. Number in none of A, B, C?}

Problem 17

Same. Only A?\text{Same. Only A?}

Problem 18

Same. Only A and B (not C)?\text{Same. Only A and B (not C)?}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

80 students; 50 Spanish, 30 French, 10 both. Only Spanish?\text{80 students; 50 Spanish, 30 French, 10 both. Only Spanish?}

Problem 20

Same. Neither language?\text{Same. Neither language?}

Problem 21

Same. At least one language?\text{Same. At least one language?}

Problem 22

100 surveyed: 60 coffee, 40 tea, 25 both. Coffee only?\text{100 surveyed: 60 coffee, 40 tea, 25 both. Coffee only?}

Problem 23

100: 70 books, 50 TV, 90 at least one. Both?\text{100: 70 books, 50 TV, 90 at least one. Both?}

Problem 24

Same as previous. Neither?\text{Same as previous. Neither?}

Problem 25

3-set scenario p11. |A∩B∩C|?\text{3-set scenario p11. |A∩B∩C|?}

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