venn Diagram

Example:

The regions in the Venn diagram below are given numbers.

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U = I, II, III, IV

A = I, II

B = II, III

A ∩ B = II

A ⋃ B = I, II, III

= III, IV

= I, IV

Example ∩

= Regions III and IV

= Regions I and IV

∩ = Region IV

To illustrate ∩ with a Venn diagram shade region IV.

Example

A survey of 100 people revealed that 60 subscribe to People magazine, 30 subscribe to Time magazine, and 20 subscribe to both magazines.

The Venn diagram below illustrates the results of the survey. The sets are labeled P for People and T for TIME.

Always start by entering the number of elements in the intersection of the sets. Twenty people subscribe to People and Time. Enter 20 in region II.

Since People has a total of 60 subscriptions, the number of elements in region I plus the number of elements in region II would equal 60. Subtract 20 from 60. This leaves 40 for region I.

Since Time has a total of 30 subscriptions, the number of elements in region II plus the number of elements in region III would equal 30. Subtract 20 from 30. This leaves 10 for region III.

To find the total number of elements in regions I, II, & III, add 40 + 20 + 10. This would equal 70. Since 100 people were surveyed, this leaves 30 for region IV.

The Venn diagram on this page can be used to answer the following questions.

1. How many are taking People or Time?

People or Time would include regions I, II, & III.

n(P ⋃ T) = 40 + 20 + 10. The total of these regions would be 70.

2. How many are taking People only?

People only would be region I.

This would be 40 subscribers.

n(P ∩ ) = 40

3. How many are taking Time only?

Time only would be region III.

This would be 10 subscribers.

n(T ∩ ) = 10

4. How many are not taking Time?

Regions not in circle T are I & IV.

Add 40 + 30. This would be 70 subscribers.

n( ) = 70

5. How many are taking neither People nor Time?

The region not in People and not in Time is region IV.

This would be 30 subscribers. n( ∩ ) = 30

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